A335037 a(n) is the number of divisors of n that are themselves divisible by the product of their digits.

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 6, 1, 3, 4, 4, 1, 5, 1, 4, 3, 3, 1, 8, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 3, 8, 1, 2, 2, 5, 1, 5, 1, 4, 5, 2, 1, 8, 2, 3, 2, 3, 1, 5, 3, 5, 2, 2, 1, 8, 1, 2, 4, 4, 2, 5, 1, 3, 2, 4, 1, 10, 1, 2, 4, 3, 3, 4, 1, 5, 3, 2, 1, 7, 2, 2, 2, 5

Offset

1,2

Comments

Inspired by A332268.

A number that is divisible by the product of its digits is called Zuckerman number (A007602); e.g., 24 is a Zuckerman number because it is divisible by 2*4=8 (see links).

a(n) = 1 iff n = 1 or n is prime not repunit >= 13.

a(n) = 2 iff n is prime = 2, 3, 5, 7 or a prime repunit.

Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 111111111111111111111 (repunit with 19 times 1's) have all divisors Zuckerman numbers. The sequence of numbers with all Zuckerman divisors is infinite iff there are infinitely many repunit primes (see A004023).

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Gérard Villemin, Nombres de Zuckerman, Types de nombres.

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A007602(n) = 3.26046... . - Amiram Eldar, Jan 01 2024

Example

For n = 4, the divisors are 1, 2, 4 and they are all Zuckerman numbers, so a(4) = 3.
For n = 14, the divisors are 1, 2, 7 and 14. Only 1, 2 and 7 are Zuckerman numbers, so a(14) = 3.

Mathematica

zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0&& Divisible[n, prodig]; a[n_] := Count[Divisors[n], _?(zuckQ[#] &)]; Array[a, 100] (* Amiram Eldar, Jun 03 2020 *)

Prog

(PARI) iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);
a(n) = sumdiv(n, d, iszu(d)); \\ Michel Marcus, Jun 03 2020

Crossrefs

Cf. A000005, A004023, A007602, A335038.

Similar with: A001227 (odd divisors), A087990 (palindromic divisors), A087991 (non-palindromic divisors), A242627 (divisors < 10), A332268 (Niven divisors).

Sequence in context: A357171 A355698 A087990 * A356069 A179940 A138707

Adjacent sequences: A335034 A335035 A335036 * A335038 A335039 A335040

Keyword

nonn,base

Author

Bernard Schott, Jun 03 2020

Status

approved