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A342572 Positive numbers all of whose prime factors are binary palindromes.

%I A342572 #30 Jul 23 2023 22:22:08
%S A342572 1,3,5,7,9,15,17,21,25,27,31,35,45,49,51,63,73,75,81,85,93,105,107,
%T A342572 119,125,127,135,147,153,155,175,189,217,219,225,243,245,255,257,279,
%U A342572 289,313,315,321,343,357,365,375,381,405,425,441,443,459,465,511,525,527
%N A342572 Positive numbers all of whose prime factors are binary palindromes.
%H A342572 Amiram Eldar, <a href="/A342572/b342572.txt">Table of n, a(n) for n = 1..10010</a> (terms below 10^7)
%F A342572 Sum_{n>=1} 1/a(n) =  Product_{p in A016041} p/(p-1) = 2.52136...
%e A342572 15 is a term since the binary representation of its prime factors, 3 and 5, are both palindromes: 11 and 101.
%e A342572 1 is a term because it has no prime factors, and "the empty set has every property". - _N. J. A. Sloane_, Jan 16 2022
%t A342572 seq[max_] := Module[{ps = Select[Range[max], PalindromeQ @ IntegerDigits[#, 2] && PrimeQ[#] &], s = {1}, s1, s2}, Do[p = ps[[k]]; emax = Floor@Log[p, max]; s1 = Join[{1}, p^Range[emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[1000]
%t A342572 Join[{1},Module[{bps=Select[Prime[Range[200]],IntegerDigits[#,2] == Reverse[ IntegerDigits[ #,2]]&]},Select[ Range[Max[ bps]],SubsetQ[ bps,FactorInteger[#][[All,1]]]&]]] (* _Harvey P. Dale_, Jan 16 2022 *)
%o A342572 (Python)
%o A342572 from sympy import factorint
%o A342572 def ispal(s): return s == s[::-1]
%o A342572 def ok(n): return n > 0 and all(ispal(bin(f)[2:]) for f in factorint(n))
%o A342572 print([k for k in range(528) if ok(k)]) # _Michael S. Branicky_, Jan 17 2022
%Y A342572 The binary version of A033620.
%Y A342572 Subsequences: A016041, A329419.
%Y A342572 Cf. A006995.
%K A342572 nonn,base
%O A342572 1,2
%A A342572 _Amiram Eldar_, Mar 27 2021
%E A342572 "Positive" added to definition by _N. J. A. Sloane_, Jan 16 2022