A337741 Numbers all of whose divisors are Niven numbers (A005349).

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

Offset: 1

Author

Amiram Eldar, Sep 17 2020

Keywords

• nonn
• base
• fini
• full

Comments

Since the only prime Niven numbers are the single-digit primes 2, 3, 5 and 7, all the terms are 7-smooth numbers (A002473).

If k is a term, all the divisors of k are also terms. Since all the terms are 7-smooth, every term is of the form p * k, where p is in {2, 3, 5, 7} and k is a smaller term. Thus it is easy to verify that there are only 31 terms in this sequence, and 1944 being the last term.

Examples

			6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Niven numbers.
		

Crossrefs

Subsequence of A002473 and A005349.

Cf. A332268, A335708, A335709.

Similar sequences: A062687, A190217, A329419.

Programs

Mathematica

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; allQ[n_] := AllTrue[Divisors[n], nivenQ]; p = {1, 2, 3, 5, 7}; s = {1}; n = 0; While[Length[s] != n, n = Length[s]; s = Select[Union @ Flatten @ Outer[Times, s, p], allQ]]; s