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

1,2

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.

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.

Example

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.

Mathematica

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

Crossrefs

Subsequence of A005349.

Sequence in context: A271955 A365420 A342650 * A342262 A255734 A357142

Adjacent sequences: A328270 A328271 A328272 * A328274 A328275 A328276

Keyword

nonn,base

Author

Amiram Eldar, Oct 10 2019

Status

approved