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

1,6

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

Example

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.

Mathematica

q[n_, b_] := (p = Times @@ IntegerDigits[n, b]) > 0 && Divisible[n, p]; a[n_] := Count[Range[2, n], _?(q[n, #] &)]; Array[a, 100]

Prog

(PARI) a(n) = sum(b=2, n-1, my(x=vecprod(digits(n, b))); x && !(n%x)); \\ Michel Marcus, Feb 12 2021

Crossrefs

Cf. A007602, A068953, A080221, A085104, A088323, A119598.

Sequence in context: A304199 A251140 A259977 * A115312 A237721 A254296

Adjacent sequences: A341431 A341432 A341433 * A341435 A341436 A341437

Keyword

nonn,base

Author

Amiram Eldar, Feb 11 2021

Status

approved