A331735 - OEIS

%I #9 Feb 15 2020 08:04:07

%S 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,

%T 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,

%U 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

%N a(n) = A009194(A225546(n)) = gcd(A225546(n), sigma(A225546(n))).

%H Antti Karttunen, <a href="/A331735/b331735.txt">Table of n, a(n) for n = 1..512</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

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

%t 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 *)

%o (PARI)

%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

%o 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))));

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

%K nonn

%O 1,8

%A _Antti Karttunen_, Feb 04 2020