A100195 -id:A100195 - cp's OEIS frontend

A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

1, 2, 2, 8, 2, 122, 2, 1682, 10082, 30242, 2, 7318082, 2, 17297282, 3632428802, 36843206402, 2, 2981705126402, 2, 1690185726028802, 3379030566912002, 28158588057602, 2, 76941821303636889602, 1077167364120207360002

Offset: 1

Vladeta Jovovic, Sep 09 2006

a(n) = 2 iff n is prime.
a(468) has 1007 decimal digits. - Michael De Vlieger, Sep 12 2018
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to row and column permutations. For example, inequivalent representatives of the a(4) = 8 matrices are:
  [1 2 3 4]
.
  [1 2] [1 2] [1 3] [1 3] [1 4] [1 4]
  [3 4] [4 3] [2 4] [4 2] [2 3] [3 2]
.
  [1]
  [2]
  [3]
  [4]
(End)
Conjecture: the sequence a(n) taken modulo a positive integer k >= 3 eventually becomes constant equal to 2. For example, the sequence taken modulo 11 is [1, 2, 2, 8, 2, 1, 2, 10, 6, 3, 2, 2, 2, 2, 2, 2, ...]. - Peter Bala, Aug 08 2025

Michael De Vlieger, Table of n, a(n) for n = 1..467
Jimmy Devillet, Gergely Kiss, Characterizations of biselective operations, arXiv:1806.02073 [math.RA], 2018.

Cf. A038041, A057625, A061095, A100195, A236696, A258899, A258903, A320444, A386877.

with(numtheory): seq(n!*add(1/(d!*(n/d)!), d in divisors(n)), n = 1..25); # Peter Bala, Aug 04 2025

f[n_] := Block[{d = Divisors@n}, Plus @@ (n!/(d! (n/d)!))]; Array[f, 25] (* Robert G. Wilson v, Sep 11 2006 *)
Table[DivisorSum[n, n!/(#!*(n/#)!) &], {n, 25}] (* Michael De Vlieger, Sep 12 2018 *)

a(n) = sumdiv(n, d, n!/(d!*(n/d)!)); \\ Michel Marcus, Sep 13 2018

E.g.f.: Sum_{k>0} (exp(x^k)-1)/k!.

More terms from Robert G. Wilson v, Sep 11 2006

A100194 Incrementally largest denominators of the Bernoulli numbers.

Original entry on oeis.org

1, 2, 6, 30, 42, 66, 2730, 14322, 1919190, 56786730, 140100870, 209191710, 2328255930, 2381714790, 7225713885390, 9538864545210, 21626561658972270, 446617991732222310, 115471236091149548610, 5145485882746933233510, 14493038256293268734790

Offset: 1

Eric W. Weisstein, Nov 08 2004

nonn

Daniel Suteu, Table of n, a(n) for n = 1..85
Eric Weisstein's World of Mathematics, Bernoulli Number
Wikipedia, Von Staudt-Clausen theorem

Cf. A100195, A000367/A002445.

Reap[For[n = record = 0, n < 1000, n = n + 2, If[(d = Denominator[BernoulliB[n]]) > record, Sow[d]; record = d]]][[2, 1]] (* Jean-François Alcover, Nov 09 2012 *)

b(n) = if((n==0) || (n>1 && n%2==1), 1, my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]+1), d[k]+1, 1))); \\ more efficient than denominator(bernfrac(n))
lista(n) = { my(m=0); for(k=0, n, my(d=b(k)); if(d>m, m=d; print1(d, ", "))); }
lista(1000); \\ Daniel Suteu, Dec 22 2018

a(21) from Seiichi Manyama, Jan 21 2017

A362285 Indices of records of A138705.

Original entry on oeis.org

0, 1, 5, 6, 8, 18, 30, 36, 90, 180, 360, 420, 504, 540, 630, 810, 840, 1080, 1260, 1680, 1890, 2520, 3240, 3780, 4200, 5040, 7560, 10080, 12600, 21420, 30240, 32760, 37800, 42840, 50400, 60480, 64260, 65520, 83160, 98280, 128520

Offset: 1

Amiram Eldar, Apr 14 2023

The corresponding record values are in A362286.

Cf. A000367, A002445, A027641, A027642, A100195, A138705, A362284, A362286.

seq[kmax_] := Module[{s = {}, mx = 0, m}, Do[m = Length[ContinuedFraction[ Abs[BernoulliB[2*k]]]]; If[m > mx, mx = m; AppendTo[s, k]], {k, 0, kmax}]; s]; seq[1000]

lista(kmax) = {my(mx = 0, m); for(k = 0, kmax, m = #contfrac(abs(bernfrac(2*k))); if(m > mx, mx = m; print1(k,", "))); }

A138705(a(n)) = A362286(n).

A362286 Record values in A138705.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 17, 23, 33, 39, 42, 46, 49, 54, 55, 57, 66, 73, 78, 83, 85, 95, 100, 105, 118, 133, 157, 162, 183, 201, 220, 224, 234, 242, 262, 272, 273, 287, 309, 314, 366

Offset: 1

Amiram Eldar, Apr 14 2023

The corresponding indices of records are in A362285.

Cf. A000367, A002445, A027641, A027642, A100195, A138705, A362284, A362285.

seq[kmax_] := Module[{s = {}, mx = 0, m}, Do[m = Length[ContinuedFraction[ Abs[BernoulliB[2*k]]]]; If[m > mx, mx = m; AppendTo[s, m]], {k, 0, kmax}]; s]; seq[1000]

lista(kmax) = {my(mx = 0, m); for(k = 0, kmax, m = #contfrac(abs(bernfrac(2*k))); if(m > mx, mx = m; print1(m,", "))); }

a(n) = A138705(A362285(n)).

cp's OEIS Frontend

A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

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A100194 Incrementally largest denominators of the Bernoulli numbers.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A362285 Indices of records of A138705.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A362286 Record values in A138705.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula