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%I #23 Sep 17 2021 17:26:17
%S 3,5,7,9,17,21,31,33,65,73,93,107,127,129,257,273,313,341,381,403,443,
%T 471,513,1025,1057,1137,1193,1273,1317,1397,1453,1571,1651,1707,1831,
%U 2047,2049,4097,4161,4321,4433,4593,4681,4841,4953,5189,5349,5461,5709
%N Positive integers whose binary representation is a palindrome and has a prime number of 1's.
%C Each term of this sequence is in both A006995 and A052294.
%H Indranil Ghosh, <a href="/A144753/b144753.txt">Table of n, a(n) for n = 1..11167</a> (terms 1..1000 from Vincenzo Librandi)
%e 21 in binary is 10101. This binary representation is a palindrome, it contains three 1's, and three is a prime. So 21 is a term.
%t okQ[n_] := Module[{idn2 = IntegerDigits[n, 2]}, (idn2 == Reverse[idn2]) && PrimeQ[First[DigitCount[n, 2]]]];Select[Range[10000], okQ] (* _Harvey P. Dale_, Sep 23 2008 *)
%o (Python)
%o from sympy import isprime
%o def ok(n): b = bin(n)[2:]; return b == b[::-1] and isprime(b.count("1"))
%o print(list(filter(ok, range(5710)))) # _Michael S. Branicky_, Sep 17 2021
%o (Python) # faster for computing initial segment of sequence
%o from sympy import isprime
%o from itertools import product
%o def ok2(bin_str): return isprime(bin_str.count("1"))
%o def bin_pals(maxdigits):
%o yield from "01"
%o digits, midrange = 2, [[""], ["0", "1"]]
%o for digits in range(2, maxdigits+1):
%o for p in product("01", repeat=digits//2-1):
%o left = "1"+"".join(p)
%o for middle in midrange[digits%2]:
%o yield left + middle + left[::-1]
%o def auptopow2(e): return [int(b, 2) for b in filter(ok2, bin_pals(e))]
%o print(auptopow2(13)) # _Michael S. Branicky_, Sep 17 2021
%Y Cf. A144752, A006995, A052294.
%K base,nonn
%O 1,1
%A _Leroy Quet_, Sep 20 2008
%E More terms from _Harvey P. Dale_, Sep 23 2008
%E Name edited by _Michael S. Branicky_, Sep 17 2021
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