A334531 Numbers that are both binary Niven numbers and binary Smith numbers.

55, 185, 205, 222, 246, 438, 623, 822, 973, 1503, 1939, 2359, 2471, 3126, 3205, 3462, 3573, 3661, 3771, 3846, 4711, 5877, 5949, 6093, 6198, 6655, 6918, 7083, 7550, 7931, 8151, 8170, 9567, 9863, 10265, 10683, 11241, 12280, 12318, 12486, 12678, 13695, 13790, 13820

Offset: 1

Author

Amiram Eldar, May 05 2020

Keywords

• nonn
• base

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Wayne L. McDaniel, On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers, Missouri Journal of Mathematical Sciences, Vol. 2, No. 3 (1990), pp. 132-136.

Crossrefs

Intersection of A049445 and A278909.

Cf. A334527.

Programs

Mathematica

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]