A372692 - OEIS

%I #7 May 10 2024 11:12:59

%S 1,1,1,3,1,1,1,1,4,1,1,3,1,1,1,15,1,4,1,3,1,1,1,1,6,1,1,3,1,1,1,5,1,1,

%T 1,12,1,1,1,1,1,1,1,3,4,1,1,15,8,6,1,3,1,1,1,1,1,1,1,3,1,1,4,3,1,1,1,

%U 3,1,1,1,4,1,1,6,3,1,1,1,15,40,1,1,3,1,1

%N The sum of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

%C The sum of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.

%C The numbers of these divisors is A372331(n) and the largest of them is A372328(n).

%H Amiram Eldar, <a href="/A372692/b372692.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = Product_{k >= 0, 2^k < e, bitand(e, 2^k) = 0} (p^(2^k) + 1).

%F a(n) >= 1, with equality if and only if n is in A036537.

%F a(n) <= n-1, with equality if and only if n = 2^(2^k) for k >= 0.

%t f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], _?(# == 0 &)])); a[1] = 1; a[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1); Array[a, 100]

%o (PARI) s(n) = apply(x -> 1 - x, binary(n));

%o a(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = s(f[i, 2]); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}

%Y Cf. A001146, A036537, A049417, A064379, A372328, A372331.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, May 10 2024