A335914 a(n) = A038040(A225546(n)).

1, 4, 12, 6, 80, 32, 2304, 24, 27, 192, 1114112, 72, 141733920768, 5120, 448, 10, 1199038364791120855040, 108, 43896425332801061786775324358698099277824, 480, 11264, 2359296, 29758566933990262223857743147232792318290386059069624958140599090033674317463552, 192, 405, 292057776128, 324, 13824

Offset

1,2

Comments

Question: Is it possible for a(n)/A331733(n) to be an integer when n is a square > 1? This is equivalent to the question whether there are odd Harmonic numbers (A001599) larger than one.

Links

Table of n, a(n) for n=1..28.

Formula

a(n) = A225546(n) * A331309(n).

Mathematica

Array[# DivisorSigma[0, #] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jul 08 2020 *)

Prog

(PARI)
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A335914(n) = if(1==n, 1, my(f=factor(n), u=#binary(vecmax(f[, 2])), prods=vector(u, x, 1), m=1, e); for(i=1, u, for(k=1, #f~, if(bitand(f[k, 2], m), prods[i] *= f[k, 1])); m<<=1); prod(i=1, u, (1+A048675(prods[i]))*(prime(i)^A048675(prods[i]))));

Crossrefs

Cf. A001599, A038040, A225546, A331309.

Sequence in context: A190392 A133517 A258565 * A377047 A331054 A372651

Adjacent sequences: A335911 A335912 A335913 * A335915 A335916 A335917

Keyword

nonn

Author

Antti Karttunen, Jul 08 2020

Status

approved