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

1, 3, 5, 7, 9, 15, 17, 21, 25, 27, 31, 35, 45, 49, 51, 63, 73, 75, 81, 85, 93, 105, 107, 119, 125, 127, 135, 147, 153, 155, 175, 189, 217, 219, 225, 243, 245, 255, 257, 279, 289, 313, 315, 321, 343, 357, 365, 375, 381, 405, 425, 441, 443, 459, 465, 511, 525, 527

Offset: 1

Author

Amiram Eldar, Mar 27 2021

Keywords

• nonn
• base

Examples

			15 is a term since the binary representation of its prime factors, 3 and 5, are both palindromes: 11 and 101.
1 is a term because it has no prime factors, and "the empty set has every property". - _N. J. A. Sloane_, Jan 16 2022
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10010 (terms below 10^7)

Crossrefs

The binary version of A033620.

Subsequences: A016041, A329419.

Cf. A006995.

Programs

Mathematica

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]
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 *)

Python

from sympy import factorint
def ispal(s): return s == s[::-1]
def ok(n): return n > 0 and all(ispal(bin(f)[2:]) for f in factorint(n))
print([k for k in range(528) if ok(k)]) # Michael S. Branicky, Jan 17 2022

Formula

Sum_{n>=1} 1/a(n) = Product_{p in A016041} p/(p-1) = 2.52136...

Extensions

"Positive" added to definition by N. J. A. Sloane, Jan 16 2022