A328273 Super Niven numbers: numbers divisible by the sums of all the nonempty subsets of their nonzero digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 24, 30, 36, 40, 48, 50, 60, 70, 80, 90, 100, 102, 110, 120, 140, 150, 200, 204, 210, 220, 240, 280, 300, 306, 330, 360, 400, 408, 420, 440, 480, 500, 510, 540, 550, 600, 630, 660, 700, 770, 800, 840, 880, 900, 990, 1000

Offset: 1

Author

Amiram Eldar, Oct 10 2019

Keywords

• nonn
• base

Comments

This sequence is infinite since if m is in the sequence then 10*m is also in the sequence.

Saadatmanesh et al. proved that:

1) The only odd terms are 1, 3, 5, 7, and 9.

2) If m is a super Niven number with k nonzero digits, then m is divisible by all the numbers 1 <= j <= k.

3) The only terms without the digit zero are 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, and 48.

Examples

			12 is in the sequence since the nonempty subsets of its nonzero digits are {1}, {2}, {1, 2}, whose sums, 1, 2, 3, are all divisors of 12.
		

References

• Majid Saadatmanesh, Super Niven numbers, MS thesis, Central Missouri State University, 1991.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Majid Saadatmanesh, Robert E. Kennedy, and Curtis Cooper, Super Niven numbers, Mathematics in College (1992), pp. 21-30.
• Amin Witno and Khaled Hyasat, Solutions to two open questions on super Niven numbers, Global Journal of Pure and Applied Mathematics, Vol. 6, No. 3 (2010), pp. 227-231, alternative link.

Crossrefs

Subsequence of A005349.

Programs

Mathematica

superNivenQ[n_] := AllTrue[Union[Total /@ Rest @ Subsets[Select[IntegerDigits[n], # > 0 &]]], Divisible[n, #] &]; Select[Range[1000], superNivenQ]