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A335037
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a(n) is the number of divisors of n that are themselves divisible by the product of their digits.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0&& Divisible[n, prodig]; a[n_] := Count[Divisors[n], _?(zuckQ[#] &)]; Array[a, 100] (* Amiram Eldar, Jun 03 2020 *)
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PROG
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(PARI) iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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