A341434 a(n) is the number of bases 1 < b < n in which n is divisible by its product of digits.
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, 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, 8, 1, 6, 3, 6, 1, 20, 2, 3, 8, 7, 3, 7, 1, 16, 7, 2, 1, 14
Offset: 1
Examples
a(3) = 1 since 3 is divisible by its product of digits only in base 2: 3 = 11_2 and 1*1 | 3. 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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
q[n_, b_] := (p = Times @@ IntegerDigits[n, b]) > 0 && Divisible[n, p]; a[n_] := Count[Range[2, n], _?(q[n, #] &)]; Array[a, 100]
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PARI
a(n) = sum(b=2, n-1, my(x=vecprod(digits(n, b))); x && !(n%x)); \\ Michel Marcus, Feb 12 2021
Formula
a(n) > 0 for all numbers n > 2 since n in base b = n-1 is 11.
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.
a(p) > 1 for primes p in A085104.
a(p) > 2 for primes p in A119598 (i.e., 31, 8191, ...).
a(n) >= A088323(n), with equality if n = 4 or if n is a prime.