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Displaying 61-70 of 309 results found. page 1 2 3 4 5 6 7 8 9 10 ... 31
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A375849 The maximum odd exponent in the prime factorization of n!. +0
2
1, 1, 3, 3, 1, 1, 7, 7, 1, 1, 5, 5, 11, 11, 15, 15, 3, 3, 1, 9, 19, 19, 3, 3, 23, 23, 25, 25, 7, 7, 31, 31, 15, 15, 17, 17, 35, 35, 9, 9, 39, 39, 41, 41, 21, 21, 3, 3, 47, 47, 49, 49, 3, 13, 53, 53, 27, 27, 9, 9, 57, 57, 63, 63, 31, 31, 31, 15, 67, 67, 11, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
COMMENTS
The sequence of indices of record values, 2, 4, 8, 14, 16, 22, 26, 28, 32, 38, ..., are the odious numbers (A000069) multiplied by 2 (A128309).
LINKS
Amiram Eldar, Table of n, a(n) for n = 2..10000
Index entries for sequences related to factorial numbers.
FORMULA
a(n) = A375032(n!).
max(a(n), A375850(n)) = A011371(n).
MATHEMATICA
a[n_] := Max[Select[FactorInteger[n!][[;; , 2]], OddQ]]; Array[a, 100, 2]
PROG
(PARI) a(n) = {my(e = select(x -> (x % 2), factor(n!)[, 2])); if(#e > 0, vecmax(e)); }
(Python)
from collections import Counter
from sympy import factorint
def A375849(n): return max(filter(lambda x: x&1, sum((Counter(factorint(i)) for i in range(2, n+1)), start=Counter()).values())) # Chai Wah Wu, Aug 31 2024
CROSSREFS
Cf. A000069, A000142, A011371, A128309, A375032, A375850.
KEYWORD
nonn,easy,new
AUTHOR
Amiram Eldar, Aug 31 2024
STATUS
approved
A375850 The maximum even exponent in the prime factorization of n!, or 0 if no such exponent exists. +0
2
0, 0, 0, 0, 0, 0, 4, 4, 2, 4, 8, 8, 10, 10, 2, 6, 6, 6, 16, 16, 18, 18, 4, 4, 22, 22, 10, 6, 6, 6, 26, 26, 14, 4, 32, 32, 34, 34, 8, 18, 38, 38, 6, 6, 6, 10, 42, 42, 46, 46, 22, 12, 12, 12, 50, 50, 26, 4, 54, 54, 56, 56, 28, 30, 30, 30, 64, 64, 66, 66, 32, 32, 70 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
The sequence of indices of record values, 0, 6, 10, 12, 18, 20, 24, 30, 34, 36, 40, ..., are the evil numbers (A001969) multiplied by 2 (A125592).
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.
FORMULA
a(n) = A375033(n!).
max(a(n), A375849(n)) = A011371(n).
MATHEMATICA
a[n_] := Max[0, Max[Select[FactorInteger[n!][[;; , 2]], EvenQ]]]; Array[a, 100, 0]
PROG
(PARI) a(n) = {my(e = select(x -> !(x % 2), factor(n!)[, 2])); if(#e == 0, 0, vecmax(e)); }
(Python)
from collections import Counter
from sympy import factorint
def A375850(n): return max(filter(lambda x: x&1^1, sum((Counter(factorint(i)) for i in range(2, n+1)), start=Counter()).values()), default=0) # Chai Wah Wu, Aug 31 2024
CROSSREFS
Cf. A000142, A001969, A011371, A125592, A375033, A375849.
KEYWORD
nonn,easy,new
AUTHOR
Amiram Eldar, Aug 31 2024
STATUS
approved
A375856 a(n) = A360706(1) XOR ... XOR A360706(n) (where XOR denotes the bitwise XOR operator). +0
2
1, 3, 0, 4, 12, 0, 5, 15, 9, 0, 7, 23, 6, 30, 16, 27, 9, 29, 16, 31, 12, 44, 8, 29, 4, 30, 8, 31, 4, 37, 0, 28, 62, 32, 61, 21, 63, 32, 96, 0, 35, 103, 65, 109, 68, 111, 72, 120, 64, 109, 44, 110, 64, 111, 44, 124, 72, 121, 64, 114, 32, 101, 4, 55, 2, 62, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
EXAMPLE
a(3) = A360706(1) XOR A360706(2) XOR A360706(3) = 1 XOR 2 XOR 3 = 0.
PROG
(PARI) { m = s = 0; for (n = 1, 67, for (v = 1, oo, if (!bittest(s, v), x = bitand(m, v); if (x==0 || x==v, s += 2^v; m = bitxor(m, v); print1 (m", "); break; ); ); ); ); }
CROSSREFS
Cf. A360706.
KEYWORD
nonn,base,new
AUTHOR
Rémy Sigrist, Aug 31 2024
STATUS
approved
A375870 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)^(3/2)) - 1) ). +0
2
1, 2, 18, 310, 8038, 280264, 12313242, 653591922, 40704551630, 2910862397646, 235114931752898, 21172206066055312, 2103333121459719446, 228525476912967164714, 26957670075375556803178, 3431314158743477432894790, 468762478424957403561956702 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..16.
FORMULA
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A349683.
a(n) = 2 * Sum_{k=0..n} (3*n+2)^(k-1) * Stirling2(n,k).
PROG
(PARI) a(n) = 2*sum(k=0, n, (3*n+2)^(k-1)*stirling(n, k, 2));
CROSSREFS
Cf. A349683, A375871, A375872.
Cf. A001861, A370056.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Sep 01 2024
STATUS
approved
A375871 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)) - 1) ). +0
2
1, 3, 30, 543, 14493, 515001, 22930869, 1229340027, 77151412902, 5551075890453, 450607640485269, 40745592546015495, 4061982705195354033, 442649982865922396337, 52351468801767526253538, 6678605910447082873015923, 914198409310749883430655441 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..16.
FORMULA
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A349683.
a(n) = 3 * Sum_{k=0..n} (3*n+3)^(k-1) * Stirling2(n,k).
PROG
(PARI) a(n) = 3*sum(k=0, n, (3*n+3)^(k-1)*stirling(n, k, 2));
CROSSREFS
Cf. A349683, A375870, A375872.
Cf. A027710, A370057.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Sep 01 2024
STATUS
approved
A375872 E.g.f. satisfies A(x) = exp( 4 * (exp(x*A(x)^(3/4)) - 1) ). +0
2
1, 4, 44, 836, 22980, 832888, 37615340, 2038258804, 128989219860, 9343113460308, 762596057754748, 69273095355130488, 6932765720797549924, 758009268677055714964, 89907747171907593677068, 11498798927333436173636612, 1577528093912610651931113908 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..16.
FORMULA
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A349683.
a(n) = 4 * Sum_{k=0..n} (3*n+4)^(k-1) * Stirling2(n,k).
PROG
(PARI) a(n) = 4*sum(k=0, n, (3*n+4)^(k-1)*stirling(n, k, 2));
CROSSREFS
Cf. A349683, A375870, A375871.
Cf. A078944, A370058.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Sep 01 2024
STATUS
approved
A375897 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/2)))^2. +0
2
1, 2, 12, 122, 1780, 34082, 810740, 23093562, 767175972, 29140904402, 1246366394548, 59292772664666, 3106206974812292, 177715679350850370, 11026719500616041076, 737552919428497318394, 52907911316906095281508, 4051998061642112552244722 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..17.
FORMULA
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052894.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (2 - exp(x))) )^2.
a(n) = (2/(n+2)!) * Sum_{k=0..n} (n+k+1)! * Stirling2(n,k).
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x)))/x)^2))
(PARI) a(n) = 2*sum(k=0, n, (n+k+1)!*stirling(n, k, 2))/(n+2)!;
CROSSREFS
Cf. A052894, A375898.
Cf. A005649.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Sep 01 2024
STATUS
approved
A375898 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/3)))^3. +0
2
1, 3, 21, 234, 3627, 72498, 1780953, 52013118, 1762754655, 68060512458, 2950869169125, 142006584810918, 7513205987292243, 433548334132153698, 27102592662130603857, 1824854382978573444174, 131676307468686605671623, 10137713081262046098901050 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..17.
FORMULA
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052894.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (2 - exp(x))) )^3.
a(n) = (3/(n+3)!) * Sum_{k=0..n} (n+k+2)! * Stirling2(n,k).
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x)))/x)^3))
(PARI) a(n) = 3*sum(k=0, n, (n+k+2)!*stirling(n, k, 2))/(n+3)!;
CROSSREFS
Cf. A052894, A375897.
Cf. A226515.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Sep 01 2024
STATUS
approved
A373694 Number of incongruent n-sided periodic Reinhardt polygons. +0
1
0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 5, 0, 1, 5, 1, 2, 10, 1, 1, 12, 4, 1, 23, 2, 1, 38, 1, 0, 64, 1, 12, 102, 1, 1, 191, 12, 1, 329, 1, 2, 633, 1, 1, 1088, 9, 34, 2057, 2, 1, 3771, 66, 12, 7156, 1, 1, 13464, 1, 1, 25503, 0, 193, 48179, 1, 2, 92206, 358, 1, 175792, 1, 1, 338202 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
LINKS
Table of n, a(n) for n=1..75.
Kevin G. Hare and Michael J. Mossinghoff, Sporadic Reinhardt Polygons, Discrete & Computational Geometry. An International Journal of Mathematics and Computer Science 49, no. 3 (2013): 540-57.
Kevin G. Hare and Michael J. Mossinghoff, Most Reinhardt Polygons Are Sporadic, Geom. Dedicata 198 (2019): 1-18.
Michael J. Mossinghoff, Enumerating Isodiametric and Isoperimetric Polygons, J. Combin. Theory Ser. A 118, no. 6 (2011): 1801-15.
Michael Mossinghoff, I love Reinhardt Polygons, ICERM 2014.
Wikipedia, Reinhardt polygon
FORMULA
a(n) = A374832(n) - A373695(n).
a(n) = Sum_{d|n, d>1} D(n/d)*Mu(2d), with D(m) = 2^floor((m-3)/2) + (Sum_{d|m, d odd} 2^(m/d)*Phi(d) )/(4m), where Mu is MoebiusMu and Phi is EulerPhi.
MATHEMATICA
dD[m_] := 2^Floor[(m - 3)/2] + Sum[2^(m/d) EulerPhi[d], {d, DeleteCases[Divisors[m], _?EvenQ]}]/4/m;
a[n_] := Sum[dD[n/d] MoebiusMu[2 d], {d, DeleteCases[Divisors[n], 1]}];
CROSSREFS
Cf. A000010, A008683, A374832, A373695.
KEYWORD
nonn,new
AUTHOR
Bernd Mulansky, Aug 04 2024
STATUS
approved
A373940 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^5). +0
1
1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 8731800, 229191600, 6352632000, 143603580120, 2736395461800, 47283190718400, 860150574738000, 20236134851478120, 614854122909391800, 19930647062659477200, 615406024970593164000, 17883373100352330768120 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Table of n, a(n) for n=0..21.
FORMULA
G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/Product_{j=1..5*k} (1 - j * x).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k).
a(n) ~ n! / (10 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 27 2024
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^5)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*stirling(j, 5, 2)*v[i-j+1])); v;
(PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2));
CROSSREFS
Cf. A000670, A052841, A353774, A353775.
Cf. A353200.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Aug 27 2024
STATUS
approved
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Last modified September 10 01:04 EDT 2024. Contains 375769 sequences. (Running on oeis4.)