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%I #14 Feb 19 2021 03:36:17
%S 0,0,1,1,1,2,2,3,2,2,1,5,2,3,4,6,1,5,1,5,4,4,1,9,2,2,4,5,1,7,3,9,4,2,
%T 3,12,1,2,3,10,1,7,2,7,7,2,1,15,2,5,3,6,1,10,3,10,4,3,1,14,1,2,7,14,3,
%U 8,1,6,3,6,1,20,2,3,8,7,3,7,1,16,7,2,1,14
%N a(n) is the number of bases 1 < b < n in which n is divisible by its product of digits.
%H Amiram Eldar, <a href="/A341434/b341434.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) > 0 for all numbers n > 2 since n in base b = n-1 is 11.
%F a(n) > 1 for all even numbers > 4 since n in base b = n-2 is 12. Similarly, a(n) > 1 for all composite numbers > 4 since if n = k*m, then n is divisible by its product of digits in bases n-m and n-k.
%F a(p) > 1 for primes p in A085104.
%F a(p) > 2 for primes p in A119598 (i.e., 31, 8191, ...).
%F a(n) >= A088323(n), with equality if n = 4 or if n is a prime.
%e a(3) = 1 since 3 is divisible by its product of digits only in base 2: 3 = 11_2 and 1*1 | 3.
%e a(6) = 2 since 6 is divisible by its product of digits in 2 bases: in base 4, 6 = 12_4 and 1*2 | 6, and in base 5, 6 = 11_5 and 1*1 | 6.
%t q[n_, b_] := (p = Times @@ IntegerDigits[n, b]) > 0 && Divisible[n, p]; a[n_] := Count[Range[2, n], _?(q[n, #] &)]; Array[a, 100]
%o (PARI) a(n) = sum(b=2, n-1, my(x=vecprod(digits(n, b))); x && !(n%x)); \\ _Michel Marcus_, Feb 12 2021
%Y Cf. A007602, A068953, A080221, A085104, A088323, A119598.
%K nonn,base
%O 1,6
%A _Amiram Eldar_, Feb 11 2021
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