Search: keyword:new
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A375257
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Numbers whose sum of base-2 digits is 1 more than their sum of base-3 digits.
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+0
0
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3, 9, 15, 28, 29, 39, 45, 57, 82, 83, 84, 85, 94, 95, 99, 110, 118, 119, 123, 135, 162, 163, 165, 174, 175, 183, 207, 219, 248, 297, 303, 315, 324, 325, 334, 335, 342, 343, 363, 382, 383, 406, 407, 411, 423, 435, 441, 447, 459, 488, 494, 496, 497, 502, 503, 506, 508, 509, 543, 570, 571, 573, 603
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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a(3) = 15 is a term because 15 = 1111_2 = 120_3 so A000120(15) = 1+1+1+1 = 4 and A053735(15) = 1+2+0 = 3.
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MAPLE
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filter:= proc(n) convert(convert(n, base, 2), `+`) = convert(convert(n, base, 3), `+`)+1 end proc:
select(filter, [$1..1000]);
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MATHEMATICA
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Select[Range[600], Subtract @@ DigitSum[#, {2, 3}] == 1 &] (* Amiram Eldar, Aug 08 2024 *)
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PROG
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(Python)
from sympy.ntheory import digits
def ok(n): return sum(digits(n, 2)[1:]) == sum(digits(n, 3)[1:]) + 1
(PARI) isok(k) = sumdigits(k, 2) == 1 + sumdigits(k, 3); \\ Michel Marcus, Aug 08 2024
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CROSSREFS
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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A374754
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a(n) is the difference between the sum of the squares and the sum of the cubes for the n first terms of A002760.
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+0
0
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0, 0, 4, -4, 5, 21, 46, 19, 55, 104, 104, 185, 285, 406, 281, 425, 594, 790, 574, 799, 1055, 1344, 1668, 1325, 1686, 2086, 2527, 3011, 2499, 3028, 3604, 4229, 4905, 4905, 5689, 6530, 7430, 8391, 7391, 8415, 9504, 10660, 11885, 13181, 11850, 13219, 14663, 16184
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OFFSET
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1,3
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COMMENTS
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For A002760(n) <= k < A002760(n+1), the difference between the sum of the squares and the sum of the cubes in the first k nonnegative integers is a(n).
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LINKS
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FORMULA
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a(1) = 0. For n >= 2, a(n) = a(n-1) + f*A002760(n) where f = 1 if A002760(n+1) is a square but not a cube, f = -1 if A002760(n) is a cube but not a square and f = 0 if A002760(n) is a square and a cube.
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EXAMPLE
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a(7) = a(6) + A002760(7) = 21 + 1*25 = 46, since 25 is a square but not a cube.
a(8) = a(7) - A002760(8) = 46 + (-1)*27 = 19, since 27 is a cube but not a square.
a(11) = a(10) + A002760(11) - A002760(11) = 104 + 0*64 = 104, since 64 is a square and a cube.
The difference between the sum of the squares and the sum of the cubes in the first 24 nonnegative integers is a(6) = 21, because A002760(6) = 16 <= 24 < A002760(7) = 25.
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MAPLE
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isA374754:=proc(k)
option remember;
if k=0 then 0
elif issqr(k) and not type(root(k, 3), integer) then procname(k-1)+k;
elif type(root(k, 3), integer) and not issqr(k) then procname(k-1)-k;
else procname(k-1)
fi;
end proc;
if k=0 then 0
elif isA374754(k)<>isA374754(k-1) or type(root(k, 6), integer) then isA374754(k)
fi;
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PROG
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(PARI) lista(nn) = my(v = select(x->issquare(x) || ispower(x, 3), [0..nn]), s=0, w = vector(#v)); for (i=1, #v, if (issquare(v[i]), s += v[i]); if (ispower(v[i], 3), s -= v[i]); w[i] = s; ); w; \\ Michel Marcus, Aug 04 2024
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CROSSREFS
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KEYWORD
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sign,new
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AUTHOR
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STATUS
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approved
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A375261
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Smallest n-digit reversible prime with only prime digits.
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+0
0
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2, 37, 337, 3257, 32233, 322573, 3222223, 32235223, 322222223, 3222222257, 32222232577, 322222232537, 3222222223333, 32222222332733, 322222222237537, 3222222222223373, 32222222222223353, 322222222222225333, 3222222222222222577, 32222222222222225573, 322222222222222233253
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OFFSET
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1,1
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LINKS
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FORMULA
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if a(n) is not a palindrome, a(n) = A177513(n) for n > 1.
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PROG
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(Python)
from sympy import isprime
from itertools import product
def a(n):
if n == 1: return 2
for first in "37":
for rest in product("2357", repeat=n-1):
s = first + "".join(rest)
if isprime(t:=int(s)) and isprime(int(s[::-1])):
return t
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CROSSREFS
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KEYWORD
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base,nonn,new
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AUTHOR
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STATUS
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approved
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A375258
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Array read by antidiagonals: T(k,n) is the least positive integer whose sum of base-2 digits is k and sum of base-3 digits is n, or -1 if there is none.
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+0
0
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1, 2, 3, -1, 6, 81, 8, 5, 28, 27, -1, 20, 7, 30, 2187, 128, 17, 14, 15, 244, 243, -1, 68, 25, 46, 31, 246, -1, 512, 8193, 26, 23, 94, 63, 6570, 19683, -1, 80, 131, 78, 47, 126, 247, 2430, 59049, 2048, 1025, 134, 53, 62, 95, 254, 255, 19926, 531441, -1, 2050, 161, 212, 79, 222, 127, 766, 2431
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OFFSET
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1,2
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COMMENTS
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T(k,n) is the least positive integer x, if it exists, such that A000120(x) = k and A053735(x) = n.
T(k,n) == n (mod 2) unless T(k,n) = -1, since A053735(x) == x (mod 2). In particular, T(1, n) = -1 if n >= 3 is odd.
Dimitrov and Howe prove that for n > 25, the sum of binary digits of 3^n is > 22. In particular, this implies T(7,1) = T(12,1) = T(21,1) = -1, since none of the first 25 powers of 3 work.
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LINKS
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EXAMPLE
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Array starts
1, 2, -1, 8, -1, 128, -1, 512, ...
3, 6, 5, 20, 17, 68, 8193, 80, ...
81, 28, 7, 14, 25, 26, 131, 134, ...
27, 30, 15, 46, 23, 78, 53, 212, ...
2187, 244, 31, 94, 47, 62, 79, 158, ...
243, 246, 63, 126, 95, 222, 125, 238, ...
-1, 6570, 247, 254, 127, 382, 223, 446, ...
19683, 2430, 255, 766, 507, 510, 383, 958, ...
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MAPLE
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T:= Matrix(8, 8, -1):
for x from 1 to 10^5 do
k:= convert(convert(x, base, 2), `+`);
n:= convert(convert(x, base, 3), `+`);
if k <= 8 and n <= 8 and T[k, n] = -1 then T[k, n]:= x; fi
od:
T;
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CROSSREFS
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KEYWORD
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sign,base,new
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AUTHOR
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STATUS
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approved
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3, 19, 103, 283, 313, 331, 463, 733, 751, 757, 1093, 1153, 1213, 1453, 1543, 1783, 2083, 2251, 2371, 2467, 2671, 2707, 2803, 3733, 3823, 7603, 7723, 8221, 9013, 9661, 14983, 15277, 15607, 16363, 16381, 16843, 17923, 19483, 20287, 21061, 22093, 23173, 24421, 24841, 25903, 27211, 28411
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OFFSET
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1,1
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COMMENTS
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Since we know the first 350199 terms of A374965, and A374965(350199) = 5026186 starts a new doubling chain, we know that any subsequent prime is greater than 5026186. This implies that the terms in the b-file, which are < 5026186, are correct. Of course, if the sequence reaches a Riesel number (cf. A076337) there will be no more primes after that point.
Note that, as can be seen from the b-file in A375028, A374965 contains many primes greater than 5026186 among the first 350199 terms, including one prime with 102410 digits. But these large primes cannot be added to the present b-file until more is discovered about primes following term 350199.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A375220
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T(n,k) is the number of permutations of the multiset {1, 1, 2, 2, ..., n, n} with k occurrences of fixed pairs (j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.
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5, 74, 15, 2193, 296, 30, 101644, 10965, 740, 50, 6840085, 609864, 32895, 1480, 75, 630985830, 47880595, 2134524, 76755, 2590, 105, 76484389121, 5047886640
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OFFSET
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2,1
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LINKS
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FORMULA
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T(n,n) = 1, T(n,n-1) = 0 (terms not in DATA),
Sum_{j=0..n-2} T(n,j) = (2*n)!/(2^n) - 1 = A000680(n) - 1,
Sum_{j=1..n-2} T(n,j) = A375223(n) - 1.
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EXAMPLE
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The triangle begins
5,
74, 15,
2193, 296, 30,
101644, 10965, 740, 50,
6840085, 609864, 32895, 1480, 75,
630985830, 47880595, 2134524, 76755, 2590, 105
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PROG
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(PARI) \\ using functions mima and a375219 from A375219, row n of triangle:
a375219(n, sizeb=2)
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CROSSREFS
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Cf. A375219 (similar for triples in the multiset).
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KEYWORD
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AUTHOR
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STATUS
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approved
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A374468
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Parity of the digit sum of n in factorial base.
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+0
0
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0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1
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OFFSET
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0
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LINKS
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FORMULA
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PROG
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(PARI)
A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
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CROSSREFS
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Characteristic function of A227149, whose complement A227148 gives the indices of 0's.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A375171
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Square array T(n,k), n>0 and k>0, read by antidiagonals in ascending order, giving the smallest n*k-digit number that, if arranged in an n X k matrix, form k-digit reversible prime in each row and n-digit reversible prime in each column, or -1 if no such number exists.
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+0
0
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2, 37, 37, 337, 1111, 337, 3257, 111331, 113131, 3257, 32233, 13139731, 113101311, 11933371, 32233, 322573, 1111179779, 113101929311, 119310213191, 1119711779, 322573, 3222223, 111111131397, 113101167919739, 1193100990013911, 111971042937997, 111119111337, 3222223
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OFFSET
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1,1
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LINKS
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FORMULA
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T(1,n) = T(n,1) <= A177513(n) for n >1.
T(1,n) = T(n,1) = A177513(n) for n = 2..6.
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EXAMPLE
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T(3,2) = 111331 is the smallest 3*2-digit number that if arranged in a 3 X 2 matrix yields in each row and column an reversible prime, i.e.,
11
13
31
-> 11 (1 time), 13 (1 time), 31 (1 time), 113 (1 time), 131 (1 time) are all reversible primes.
Table begins (upper left corner = T(1,1)):
2 37 337 3257 ...
37 1111 113131 11933371 ...
337 111331 113101311 119310213191 ...
3257 13139731 113101929311 1193100990013911 ...
... ... ... ... ...
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PROG
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(PARI)
isp(x) = ispseudoprime(x) && ispseudoprime(fromdigits(Vecrev(digits((x)))));
ispd(x) = ispseudoprime(fromdigits(x)) && ispseudoprime(fromdigits(Vecrev(x)));
vp(n) = select(isp, [10^(n-1)..10^n-1]);
isok(val, n, k) = my(d=digits(val), v=vector(k, i, []), j=1); for (i=1, #d, v[j] = concat(v[j], d[i]); j++; if (j>k, j=1); ); for (i=1, k, if (!ispd(v[i]), return(0)); ); return(1);
T(n, k) = my(v = vp(k), nbp = #v, nb = nbp^n); for (i=0, nb-1, my(d=digits(i, nbp)); if (d==[], d=vector(n)); while(#d <n, d=concat(0, d)); d = apply(x->x+1, d); my(s=""); for (i=1, #d, s = concat(s, Str(v[d[i]]))); my(val = eval(s)); if (isok(val, n, k), return(val)); ); \\ Michel Marcus, Aug 08 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A375245
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Number of biquadratefree numbers <= n.
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+0
0
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61, 62, 63
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{d>=1} mu(d)*floor(n/d^4), where mu is the Moebius function A008683.
n/a(n) converges to zeta(4).
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PROG
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(Python)
from sympy import mobius, integer_nthroot
def A375245(n): return int(sum(mobius(k)*(n//k**4) for k in range(1, integer_nthroot(n, 4)[0]+1)))
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CROSSREFS
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KEYWORD
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nonn,easy,new
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AUTHOR
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STATUS
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approved
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A375219
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T(n,k) is the number of permutations of the multiset {1, 1, 1, 2, 2, 2, ..., n, n, n} with k occurrences of fixed triples (j,j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.
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+0
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OFFSET
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2,1
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COMMENTS
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Trivially, T(n,n) = 1 and T(n,n-1) = 0.
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LINKS
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FORMULA
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Sum_{j=0..n-2} T(n,j) = (3*n)!/(6^n) - 1 = A014606(n) - 1.
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EXAMPLE
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The triangle begins
19,
1622, 57,
362997, 6488, 114,
166336604, 1814985, 16220, 190
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T(2,0) = 19: the permutations of {1,1,1,2,2,2} with no fixed triples are
[1,1,2,1,2,2], [1,1,2,2,1,2], [1,1,2,2,2,1], [1,2,1,1,2,2], [1,2,1,2,1,2], [1,2,1,2,2,1], [1,2,2,1,1,2], [1,2,2,1,2,1], [1,2,2,2,1,1], [2,1,1,1,2,2], [2,1,1,2,1,2], [2,1,1,2,2,1], [2,1,2,1,1,2], [2,1,2,1,2,1], [2,1,2,2,1,1], [2,2,1,1,1,2], [2,2,1,1,2,1], [2,2,1,2,1,1], [2,2,2,1,1,1].
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PROG
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(PARI) mima (x, n1=1, i2=-oo) = {my (n2, n=#x, mi=x[n1], ma=mi); n2=if (i2<=0, n, min(n, i2)); for (i=n1+1, n2, if (x[i]<mi, mi=x[i], if (x[i]>ma, ma=x[i]))); [mi, ma]};
\\ returns row n of triangle, bsize is the block size in the multiset.
a375219(n, bsize=3) = {my (p=vector(bsize*n, i, 1+(i-1)\bsize), r=s=vector(n), m=vector(n-1)); forperm (p, q, for (b=1, n, my (bm=bsize*(b-1), j=mima(q, bm+1, bm+bsize)); r[b]=j[1]; s[b]=j[2]); my (rs=vector(n, i, r[i]==i && s[i]==i)); for (k=0 , n-2, m[k+1]+=vecsum(rs)==k)); m}
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CROSSREFS
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Cf. A374980, A375223 (columns 0 and 1 in a similar triangle for the multiset {1, 1, 2, 2, ..., n, n}).
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KEYWORD
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AUTHOR
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STATUS
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approved
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