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a(n) is the number of divisors of n that have only prime-indexed prime factors.
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%I #16 Feb 04 2023 14:14:44

%S 1,1,2,1,2,2,1,1,3,2,2,2,1,1,4,1,2,3,1,2,2,2,1,2,3,1,4,1,1,4,2,1,4,2,

%T 2,3,1,1,2,2,2,2,1,2,6,1,1,2,1,3,4,1,1,4,4,1,2,1,2,4,1,2,3,1,2,4,2,2,

%U 2,2,1,3,1,1,6,1,2,2,1,2,5,2,2,2,4,1,2

%N a(n) is the number of divisors of n that have only prime-indexed prime factors.

%C First differs from A322976 at n = 21.

%C Equivalently, a(n) is the number of divisors of the largest divisor of n that has only prime-indexed prime factors.

%C The asymptotic mean of this sequence is Product_{p in A006450} p/(p-1) > 3. See A076610 for a numerical estimate of the value of this product.

%H Amiram Eldar, <a href="/A360326/b360326.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 if and only if n is in A320628.

%F a(n) = A000005(n) if and only if n is in A076610.

%F a(n) = A000005(A360325(n)).

%F Multiplicative with a(p^e) = e+1 if p is a prime-indexed prime (A006450), and 1 otherwise.

%t f[p_, e_] := If[PrimeQ[PrimePi[p]], e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), e[i]+1, 1));}

%Y Cf. A000005, A006450, A076610, A320628, A322976, A360325, A360327.

%K nonn,mult

%O 1,3

%A _Amiram Eldar_, Feb 03 2023