A328273 - OEIS

%I #9 Oct 12 2019 03:37:35

%S 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,

%T 120,140,150,200,204,210,220,240,280,300,306,330,360,400,408,420,440,

%U 480,500,510,540,550,600,630,660,700,770,800,840,880,900,990,1000

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

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

%C Saadatmanesh et al. proved that:

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

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

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

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

%H Giovanni Resta, <a href="/A328273/b328273.txt">Table of n, a(n) for n = 1..10000</a>

%H Majid Saadatmanesh, Robert E. Kennedy, and Curtis Cooper, <a href="http://cs.ucmo.edu/~cnc8851/articles/super.pdf">Super Niven numbers</a>, Mathematics in College (1992), pp. 21-30.

%H Amin Witno and Khaled Hyasat, <a href="http://www.ripublication.com/Volume/gjpamv6n3.htm">Solutions to two open questions on super Niven numbers</a>, Global Journal of Pure and Applied Mathematics, Vol. 6, No. 3 (2010), pp. 227-231, <a href="https://www.thefreelibrary.com/Solutions+to+two+open+questions+on+super+Niven+numbers.-a0323038713">alternative link</a>.

%e 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.

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

%Y Subsequence of A005349.

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Oct 10 2019