A069248 Number of positive divisors of n themselves divisible by largest prime that divides n.

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

Offset

1,4

Links

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Formula

a(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 largest prime to divide n in the prime factorization of n.

Example

The divisors of 12 which are themselves divisible by 3 (the largest prime dividing 12) are 3, 6 and 12. So the 12th term is 3.

Mathematica

nd[n_]:=Module[{lp=FactorInteger[n][[-1, 1]]}, Count[Divisors[n], _?(Mod[ #, lp] == 0&)]]; Join[{0}, Array[nd, 100, 2]] (* Harvey P. Dale, May 05 2019 *)
a[n_] := Last[e = FactorInteger[n][[;; , 2]]]*Times @@ (1 + Most[e]); a[1] = 0; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

Prog

(PARI) a(n) = if (n==1, 0, gp = vecmax(factor(n)[, 1]); sumdiv(n, d, ((d%gp) == 0))); \\ Michel Marcus, Feb 10 2017

Crossrefs

Cf. A000005, A071178.

Sequence in context: A324190 A098893 A302037 * A329378 A329617 A008481

Adjacent sequences: A069245 A069246 A069247 * A069249 A069250 A069251

Keyword

nonn

Author

Leroy Quet, Apr 08 2002

Status

approved