A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

1, 1, 2, 1, 6, 2, 24, 1, 1, 6, 120, 2, 720, 24, 3, 1, 5040, 1, 40320, 6, 24, 120, 362880, 2, 3, 720, 2, 24, 3628800, 3, 39916800, 1, 120, 5040, 24, 1, 479001600, 40320, 720, 6, 6227020800, 24, 87178291200, 120, 6, 362880, 1307674368000, 2, 12, 3, 5040, 720

Offset: 1

Author

Gus Wiseman, May 18 2019

Keywords

• nonn
• frac
• look

Comments

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the denominator of 1/i_1! + ... + 1/i_k!.

Links

• Robert Israel, Table of n, a(n) for n = 1..3168
• Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Crossrefs

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.

Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325704.

Programs

Maple

f:= proc(n) local F,t;
    F:= ifactors(n)[2];
    denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024

Mathematica

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator

Formula

a(n) = A318574(A325709(n)).