A069157 Number of positive divisors of n that are divisible by the smallest prime that divides n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 4, 1, 5, 2, 2, 2, 6, 1, 2, 2, 6, 1, 4, 1, 4, 4, 2, 1, 8, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 4, 6, 2, 4, 1, 4, 2, 4, 1, 9, 1, 2, 3, 4, 2, 4, 1, 8, 4, 2, 1, 8, 2, 2, 2, 6, 1, 6, 2, 4, 2, 2, 2, 10, 1, 3, 4, 6

Offset: 1

Author

Leroy Quet, Apr 08 2002

Keywords

• nonn
• easy

Examples

			The divisors of 12 which are themselves divisible by 2 (the smallest prime dividing 12) are 2, 4, 6 and 12. So the 12th term is 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A020639, A067029.

Programs

Mathematica

a[1] = 0; a[n_] := DivisorSigma[0, n] * (e = FactorInteger[n][[1, 2]])/(e + 1); Array[a, 100] (* Amiram Eldar, May 06 2020 *)

PARI

a(n) = if (n==1, 0, my(p=vecmin(factor(n)[,1])); sumdiv(n, d, ((d % p) == 0))); \\ Michel Marcus, May 06 2020

Python

from sympy import divisor_count, factorint
def a067029(n): return 0 if n==1 else next(iter(factorint(n).values()))
def a(n): return divisor_count(n)*a067029(n)//(1 + a067029(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 12 2017

Scheme

(define (A069157 n) (let ((e_n (A067029 n))) (* (/ e_n (+ 1 e_n)) (A000005 n)))) ;; (After the formula given by the author of the sequence) - Antti Karttunen, Aug 12 2017

Formula

a(n) = A000005(n) * A067029(n)/(1+A067029(n)) = d(n) * e_n/(e_n + 1), where d(n) is the number of positive divisors of n and e_n is the exponent of the smallest prime to divide n in the prime factorization of n.

a(p) = 1 iff p is prime. - Bernard Schott, May 06 2020

a(n) = A000005(n/p) where p is the smallest prime dividing n. - David A. Corneth, May 06 2020