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

Author

Antti Karttunen, Jul 08 2020

Keywords

• nonn

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.

Crossrefs

Cf. A001599, A038040, A225546, A331309.

Programs

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 *)

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]))));

Formula

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