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

1,2

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

(Magma) [#[d:d in Divisors(k)|d mod &+Intseq(d) eq 0]:k  in [1..100]];
(PARI) a(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ Michel Marcus, May 04 2020

Crossrefs

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

Sequence in context: A138707 A358099 A095048 * A355593 A355302 A084302

Adjacent sequences: A332265 A332266 A332267 * A332269 A332270 A332271

Keyword

nonn,base

Author

Marius A. Burtea, May 04 2020

Status

approved