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

1,4

Links

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A020639, A067029.

Sequence in context: A039637 A194548 A274009 * A294894 A076526 A351417

Adjacent sequences: A069154 A069155 A069156 * A069158 A069159 A069160

Keyword

nonn,easy

Author

Leroy Quet, Apr 08 2002

Status

approved