A331735 a(n) = A009194(A225546(n)) = gcd(A225546(n), sigma(A225546(n))).

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 3, 1, 12, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 12, 1, 9, 1, 4, 1, 1, 1, 10, 1, 3, 1, 1, 1, 1, 1, 12, 1

Offset: 1

Author

Antti Karttunen, Feb 04 2020

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..512
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A009194, A225546, A331733, A331734.

Programs

Mathematica

Array[GCD[#, DivisorSigma[1, #]] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Feb 12 2020 *)

PARI

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331735(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); gcd(prod(i=1,u,prime(i)^A048675(prods[i])), prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1))));

Formula

a(n) = A009194(A225546(n)) = gcd(A225546(n), A331733(n)).