1,3

a(n) is the number of positive integers k with A000005(k) = n and A006530(k) <= n.

By the divisor function and the fundamental theorem of arithmetic, the triangle T(n,k) in A251683 gives the number of positive integers with n divisors and k given prime factors; since these k primes can be chosen from those <= n, in the formula section T(n,k) is weighted by binomial(pi(n), k).

This sequence extends the prime counting function to all integers. On a prime p, a(p) matches pi(p) (the number of primes up to p). For prime-indexed primes (A006450), the values are themselves prime; in fact a(p)=k when p is the k-th prime. Proofs see formula section.

Felix Huber, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Divisor Function

Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic

a(n) = Sum_{k=1..A086436(n)} T(n,k)*binomial(A000720(n), k), where T(n,k) is the triangle in A251683.

a(p) = A000720(p) for primes p. Proof: A086436(p) = 1 and T(p,1) = 1 for primes p, so a(n) = 1*binomial(A000720(p), 1) = A000720(p). Equivalently, A390375(A000040(k)) = k for positive integers k.

a(p) is prime iff a prime p is in A006450. Proof: By the definition of A006450, a prime p belongs to A006450 iff pi(p) is prime. Therefore a(p) is prime <=> pi(p) is prime <=> p is in A006450.

a(6) = 9 because 12, 18, 20, 32, 45, 50, 75, 243 and 3125 are the only 9 positive integers with 6 divisors and no prime divisor greater than 6.

a(13) = 6 because 13 is the sixth prime number.

# b and t by Alois P. Heinz (A251683)

with(numtheory):

b:=proc(n) option remember;

expand(x*(1+add(b(n/d), d=divisors(n) minus {1, n})))

end:

t:=n->(p->[seq(coeff(p, x, i), i=1..degree(p))])(b(n)):

A390375:=proc(n)

T:=t(n);

return max(1, add(T[k]*binomial(pi(n), k), k=1..nops(T)));

end proc:

seq(A390375(n), n=1..58);

b[n_]:=x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]); T[n_]:=CoefficientList[b[n]/x, x];

a[n_]:=If[n>1, Sum[T[n][[k]]*Binomial[PrimePi[n], k], {k, 1, PrimeOmega[n]}], 1]; Array[a, 58] (* James C. McMahon, Nov 08 2025 *)

(Python)

from sympy.functions.combinatorial.numbers import primepi

def A390375(n: int) -> int:

ordfact = A251683_row(n)

return max(1, sum(f * binomial(primepi(n), k + 1) for k, f in enumerate(ordfact)))

print([A390375(n) for n in range(1, 59)]) # Peter Luschny, Nov 06 2025

Cf. A000005, A000720, A005179, A006450, A006530, A007318, A027750, A074206, A086436, A251683, A377505, A377537.

Sequence in context: A059083 A207626 A232324 * A124931 A210226 A209163

Adjacent sequences: A390372 A390373 A390374 * A390376 A390377 A390378

nonn,easy

Felix Huber, Nov 05 2025

approved