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

1,4

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

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)

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000005, A077761, A010051, A355937.

Sequence in context: A317751 A106490 A349281 * A327399 A122375 A038548

Adjacent sequences: A345219 A345220 A345221 * A345223 A345224 A345225

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 11 2021

Status

approved