A163411 A positive integer is included if it is a palindrome when written in binary, and it is divisible by at least one prime that is not a binary palindrome.
33, 65, 99, 129, 165, 195, 231, 273, 297, 325, 341, 387, 403, 427, 455, 471, 495, 513, 561, 585, 633, 645, 693, 717, 819, 843, 891, 903, 951, 975, 1023, 1025, 1057, 1105, 1137, 1161, 1273, 1317, 1365, 1397, 1421, 1501, 1539, 1651, 1675, 1707
Offset: 1
Examples
99 in binary is 1100011, which is a palindrome. 99 is divisible by the primes 3 and 11. 3 in binary is 11, a palindrome. But 11(decimal) in binary is 1011, not a palindrome. Since there is at least one prime dividing the binary palindrome 99 that is not a binary palindrome, then 99 is included in this sequence.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
dmax:= 15: # to get all terms with at most dmax binary digits revdigs:= proc(n) local L, Ln, i; L:= convert(n, base, 2); Ln:= nops(L); add(L[i]*2^(Ln-i), i=1..Ln); end proc: isbpali:= proc(n) option remember; local L; L:= convert(n,base,2); L=ListTools:-Reverse(L) end proc: Bp:= {0, 1}: for d from 2 to dmax do if d::even then Bp:= Bp union {seq(2^(d/2)*x + revdigs(x), x=2^(d/2-1)..2^(d/2)-1)} else m:= (d-1)/2; B:={seq(2^(m+1)*x + revdigs(x), x=2^(m-1)..2^m-1)}; Bp:= Bp union B union map(`+`, B, 2^m) fi od: R:= select(t -> ormap(not isbpali, numtheory:-factorset(t)), Bp): sort(convert(R,list)); # Robert Israel, Dec 19 2016 -
Mathematica
a = {}; For[n = 2, n < 10000, n++, If[FromDigits[Reverse[IntegerDigits[n, 2]], 2] == n, b = Table[FactorInteger[n][[i, 1]], {i, 1, Length[FactorInteger[n]]}]; For[i = 1, i < Length[b] + 1, i++, If[ ! FromDigits[Reverse[IntegerDigits[b[[i]], 2]], 2] == b[[i]], AppendTo[a, n]; Break]]]]; a (* Stefan Steinerberger, Aug 05 2009 *)
Extensions
More terms from Stefan Steinerberger, Aug 05 2009
Corrected by Leroy Quet, Aug 09 2009
Comments