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Number of divisors of n with relatively prime prime indices, meaning they belong to A289509.
3

%I #16 Dec 06 2021 03:06:49

%S 0,1,0,2,0,2,0,3,0,2,0,4,0,2,1,4,0,3,0,4,0,2,0,6,0,2,0,4,0,5,0,5,1,2,

%T 1,6,0,2,0,6,0,4,0,4,2,2,0,8,0,3,1,4,0,4,1,6,0,2,0,9,0,2,0,6,0,5,0,4,

%U 1,5,0,9,0,2,2,4,1,4,0,8,0,2,0,8,1,2,0

%N Number of divisors of n with relatively prime prime indices, meaning they belong to A289509.

%C A prime index of n is a number m such that prime(m) divides n.

%H Antti Karttunen, <a href="/A318979/b318979.txt">Table of n, a(n) for n = 1..20000</a>

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

%F a(n) = A000005(n) - A327657(n). - _Antti Karttunen_, Dec 05 2021

%e The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, corresponding to the prime index multisets (), (1), (2), (11), (12), (22), (112), (122), (1122) respectively. Of these, only (1), (11), (12), (112), (122), (1122) are relatively prime, corresponding to the divisors 2, 4, 6, 12, 18, 36, so a(36) = 6.

%t Table[Length[Select[Divisors[n],GCD@@PrimePi/@FactorInteger[#][[All,1]]==1&]],{n,100}]

%o (PARI) a(n) = sumdiv(n, d, gcd(apply(x->primepi(x), factor(d)[,1])) == 1); \\ _Michel Marcus_, Jan 09 2019

%Y Cf. A000005, A000837, A001221, A018783, A056239, A289508, A289509, A296150, A298748, A318978, A327657.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 06 2018