A332268 a(n) is the number of divisors of n that are Niven numbers.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 4, 2, 1, 8, 2, 2, 4, 4, 1, 7, 1, 4, 2, 2, 3, 9, 1, 2, 2, 8, 1, 7, 1, 3, 5, 2, 1, 9, 2, 5, 2, 3, 1, 8, 2, 5, 2, 2, 1, 11, 1, 2, 6, 4, 2, 4, 1, 3, 2, 6, 1, 12, 1, 2, 3, 3, 2, 4, 1, 9, 5, 2, 1, 10, 2, 2

Offset: 1

Author

Marius A. Burtea, May 04 2020

Keywords

• nonn
• base

Comments

If p is a prime number, p >= 11, then a(p) = 1.

Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 36, 40, 54, 63, 72, 81, 108, 162, 216, 243, 324, 486, 648, 972, 1944, have all divisors Niven numbers. There are only finitely many numbers all of whose divisors are Niven numbers. (A337741).

A333456(n) is the least number k such that a(k) = n. - Bernard Schott, Jul 30 2022

Examples

			For n = 4 the divisors are 1, 2, 4 and they are all Niven numbers, so a(4) = 3.
For n = 14 the divisors are 1, 2, 7 and 14. Only 1, 2 and 7 are Niven numbers, so a(14) = 3.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A005349, A333456, A337741, A340637.

Programs

Magma

[#[d:d in Divisors(k)|d mod &+Intseq(d) eq 0]:k  in [1..100]];

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; Array[a, 100] (* Amiram Eldar, May 04 2020 *)

PARI

a(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ Michel Marcus, May 04 2020

Formula

a(A333456(n)) = n. - Bernard Schott, Jul 30 2022