A069157 - OEIS

%I #35 May 06 2020 05:10:43

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A069157/b069157.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

%F a(p) = 1 iff p is prime. - _Bernard Schott_, May 06 2020

%F a(n) = A000005(n/p) where p is the smallest prime dividing n. - _David A. Corneth_, May 06 2020

%e 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.

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

%o (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

%o (Python)

%o from sympy import divisor_count, factorint

%o def a067029(n): return 0 if n==1 else next(iter(factorint(n).values()))

%o def a(n): return divisor_count(n)*a067029(n)//(1 + a067029(n))

%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Aug 12 2017

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

%Y Cf. A000005, A020639, A067029.

%K nonn,easy

%O 1,4

%A _Leroy Quet_, Apr 08 2002