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

%I A335037 #44 Jan 01 2024 08:00:44
%S A335037 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,
%T A335037 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,
%U A335037 2,4,1,10,1,2,4,3,3,4,1,5,3,2,1,7,2,2,2,5
%N A335037 a(n) is the number of divisors of n that are themselves divisible by the product of their digits.
%C A335037 Inspired by A332268.
%C A335037 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).
%C A335037 a(n) = 1 iff n = 1 or n is prime not repunit >= 13.
%C A335037 a(n) = 2 iff n is prime = 2, 3, 5, 7 or a prime repunit.
%C A335037 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).
%H A335037 David A. Corneth, <a href="/A335037/b335037.txt">Table of n, a(n) for n = 1..10000</a>
%H A335037 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/Zuckerman_number/">Zuckerman numbers</a>, Numbers Aplenty.
%H A335037 Gérard Villemin, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Zuckerma.htm#Zuck">Nombres de Zuckerman</a>, Types de nombres.
%F A335037 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
%e A335037 For n = 4, the divisors are 1, 2, 4 and they are all Zuckerman numbers, so a(4) = 3.
%e A335037 For n = 14, the divisors are 1, 2, 7 and 14. Only 1, 2 and 7 are Zuckerman numbers, so a(14) = 3.
%t A335037 zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0&& Divisible[n, prodig]; a[n_] := Count[Divisors[n], _?(zuckQ[#] &)]; Array[a, 100] (* _Amiram Eldar_, Jun 03 2020 *)
%o A335037 (PARI) iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);
%o A335037 a(n) = sumdiv(n, d, iszu(d)); \\ _Michel Marcus_, Jun 03 2020
%Y A335037 Cf. A000005, A004023, A007602, A335038.
%Y A335037 Similar with: A001227 (odd divisors), A087990 (palindromic divisors), A087991 (non-palindromic divisors), A242627 (divisors < 10), A332268 (Niven divisors).
%K A335037 nonn,base
%O A335037 1,2
%A A335037 _Bernard Schott_, Jun 03 2020