A334531 - OEIS

%I #11 May 06 2020 01:46:36

%S 55,185,205,222,246,438,623,822,973,1503,1939,2359,2471,3126,3205,

%T 3462,3573,3661,3771,3846,4711,5877,5949,6093,6198,6655,6918,7083,

%U 7550,7931,8151,8170,9567,9863,10265,10683,11241,12280,12318,12486,12678,13695,13790,13820

%N Numbers that are both binary Niven numbers and binary Smith numbers.

%H Amiram Eldar, <a href="/A334531/b334531.txt">Table of n, a(n) for n = 1..10000</a>

%H Wayne L. McDaniel, <a href="https://doi.org/10.35834/1990/0203132">On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers</a>, Missouri Journal of Mathematical Sciences, Vol. 2, No. 3 (1990), pp. 132-136.

%e The binary representation of 55 is 110111. It is a binary Niven number since 1 + 1 + 0 + 1 + 1 + 1 = 5 is a divisor of 55. It is also a binary Smith number since its prime factorization, 5 * 11, is 101 * 1011 in binary representation, and 1 + 1 + 0 + 1 + 1 + 1 = (1 + 0 + 1) + (1 + 0 + 1 + 1). Thus 55 is a term.

%t binWt[n_] := DigitCount[n, 2, 1]; binNivenSmithQ[n_] := Divisible[n, (bw = binWt[n])] && CompositeQ[n] && Plus @@ (Last@# * binWt[First@#] & /@ FactorInteger[n]) == bw; Select[Range[10^4], binNivenSmithQ]

%Y Intersection of A049445 and A278909.

%Y Cf. A334527.

%K nonn,base

%O 1,1

%A _Amiram Eldar_, May 05 2020