A334532 Binary palindromic numbers that are also binary Niven and binary Smith numbers.
22517, 317273, 5876429, 7129499, 18659953, 20053785, 24328605, 28676955, 31134135, 88700053, 92254197, 95682157, 96316909, 97462173, 117812487, 120026919, 120303271, 120323751, 128167471, 133396095, 133984767, 292610513, 309416393, 314572713, 348580965, 351400421
Offset: 1
Examples
The binary representation of 22517 is 101011111110101 which is palindromic. The number of 1's in its binary representation is 11 which is a divisor of 22517, hence 22517 is a binary Niven. It is also a binary Smith number since its prime factorization, 11 * 23 * 89, is 1011 * 10111 * 1011001 in binary representation, and (1 + 0 + 1 + 1) + (1 + 0 + 1 + 1 + 1) + (1 + 0 + 1 + 1 + 0 + 0 + 1) = 3 + 4 + 4 = 11 is equal to the number of 1's in its binary representation.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..779
- Amin Witno, Smith Numbers With Extra Digital Features, Integers, Vol. 14 (2014), Article A66.
Crossrefs
Programs
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Mathematica
binWt[n_] := DigitCount[n, 2, 1]; binPalNivenSmithQ[n_] := Divisible[n, (bw = Plus @@ (d = IntegerDigits[n, 2]))] && PalindromeQ[d] && CompositeQ[n] && Plus @@ (Last@# * binWt[First@#] & /@ FactorInteger[n]) == bw; Select[Range[2*10^6], binPalNivenSmithQ]