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%I #10 May 06 2020 01:46:32
%S 22517,317273,5876429,7129499,18659953,20053785,24328605,28676955,
%T 31134135,88700053,92254197,95682157,96316909,97462173,117812487,
%U 120026919,120303271,120323751,128167471,133396095,133984767,292610513,309416393,314572713,348580965,351400421
%N Binary palindromic numbers that are also binary Niven and binary Smith numbers.
%H Amiram Eldar, <a href="/A334532/b334532.txt">Table of n, a(n) for n = 1..779</a>
%H Amin Witno, <a href="https://www.emis.de/journals/INTEGERS/papers/o66/o66.Abstract.html">Smith Numbers With Extra Digital Features</a>, Integers, Vol. 14 (2014), Article A66.
%e 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.
%t 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]
%Y Intersection of A006995, A049445 and A278909.
%Y Intersection of any two of the sequences A334529, A334530 and A334531.
%Y Cf. A334528.
%K nonn,base
%O 1,1
%A _Amiram Eldar_, May 05 2020
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