A336997 - OEIS

%I #12 Apr 18 2026 16:53:45

%S 1,4,10,56,136,1952,5104,94208,605056,7741952,39917824,1458295808,

%T 6227024896,175463616512,2353813878784,48886264659968,355687428161536,

%U 17362063156969472,121645100409094144,6001501553433509888,85800344155030552576,2248030289949388439552

%N a(n) = n! * Sum_{d|n} 2^(d - 1) / d!.

%H Robert Israel, <a href="/A336997/b336997.txt">Table of n, a(n) for n = 1..448</a>

%F E.g.f.: Sum_{k>=1} (exp(2*x^k) - 1) / 2.

%F a(p) = p! + 2^(p - 1), where p is prime.

%p f:= proc(n) local d;  n! * add(2^(d-1)/d!, d = NumberTheory:-Divisors(n)) end proc:

%p map(f, [$1..25]); # _Robert Israel_, Apr 16 2026

%t Table[n! Sum[2^(d - 1)/d!, {d, Divisors[n]}], {n, 1, 22}]

%t nmax = 22; CoefficientList[Series[Sum[(Exp[2 x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%o (PARI) a(n) = n! * sumdiv(n, d, 2^(d-1)/d!); \\ _Michel Marcus_, Aug 12 2020

%Y Cf. A010842, A034729, A057625, A336998.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Aug 10 2020