A345222 Number of divisors of n with a prime number of divisors.

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 3, 2

Offset: 1

Author

Wesley Ivan Hurt, Jun 11 2021

Keywords

• nonn
• easy

Comments

Number of divisors of n that are powers of primes with an exponent k such that k+1 is a prime. - Amiram Eldar, Oct 06 2023

Inverse Möbius transform of sequence b(1) = 0, b(n) = A355937(n) for n > 1, or equivalently, one less than the inverse Möbius transform of A355937. - Antti Karttunen, Oct 06 2023

Examples

			a(12) = 3; The divisors of 12 are {1, 2, 3, 4, 6, 12} and the corresponding number of divisors of each of these numbers is {1, 2, 2, 3, 4, 6}. Thus, there are 3 divisors of 12 with a prime number of divisors.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000005, A077761, A010051, A355937.

Programs

Mathematica

Table[Sum[(PrimePi[DivisorSigma[0, k]] - PrimePi[DivisorSigma[0, k] - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
f[p_, e_] := PrimePi[e+1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a,100] (* Amiram Eldar, Oct 06 2023 *)

PARI

a(n) = sumdiv(n, d, isprime(numdiv(d))); \\ Michel Marcus, Jun 11 2021

Formula

a(n) = Sum_{d|n} c(tau(d)), where c(n) is the prime characteristic.

From Amiram Eldar, Oct 06 2023: (Start)

Additive with a(p^e) = primepi(e+1).

Sum_{k=1..n} a(k) ~ n * (log(n) + B + C), where B is Mertens's constant (A077761), and C = Sum_{k>=2} P(prime(k)-1) = 0.54756961912815344341..., where P(s) is the prime zeta function. (End)