A331741 -id:A331741 - cp's OEIS frontend

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

1, 3, 7, 4, 31, 15, 511, 12, 13, 63, 131071, 28, 8589934591, 1023, 127, 6, 36893488147419103231, 39, 680564733841876926926749214863536422911, 124, 2047, 262143, 231584178474632390847141970017375815706539969331281128078915168015826259279871, 60, 121, 17179869183, 91, 2044

Offset: 1

Antti Karttunen, Feb 02 2020

nonn

Cf. A000203, A048675, A225546, A331309, A331734, A331735, A331741.

Cf. A323243, A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

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

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331733(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,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A000203(A225546(n)).

For all n >= 1, A000035(a(A016754(n))) = 1. [Result is odd for all odd squares]

A331734 a(n) = A033879(A225546(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, 5, 1, 1, -4, 1, 1, 1, 4, 1, -3, 1, -28, 1, 1, 1, -12, 41, 1, -19, -508, 1, 1, 1, 2, 1, 1, 1, 14, 1, 1, 1, -60, 1, 1, 1, -131068, -115, 1, 1, -2, 3281, -39, 1, -8589934588, 1, -51, 1, -1020, 1, 1, 1, -124, 1, 1, -2035, 6, 1, 1, 1, -36893488147419103228, 1, 1, 1, -12, 1, 1, -199, -680564733841876926926749214863536422908

Offset: 1

Antti Karttunen, Feb 02 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..136

Cf. A005117, A033879, A225546, A331733, A331735, A331741.

Cf. A323244, A323174, A324055, A324185, A324546 for other permutations of the deficiency, and also A324574, A324654.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331734(n) = if(issquarefree(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); (2*prod(i=1,u,prime(i)^A048675(prods[i]))) - prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A033879(A225546(n)) = 2*A225546(n) - A331733(n).
For all n, a(A005117(n)) = 1. [It is not known if there are 1's in any other positions. See Jianing Song's Oct 13 2019 comment in A033879.]
For a necessary condition that a(s) would be zero for any square, see A331741.

cp's OEIS Frontend

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

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