A365451 Odd composite numbers k such that A349494(k) = A000120(k).

15, 27, 51, 63, 85, 95, 111, 119, 123, 125, 187, 219, 221, 255, 335, 365, 411, 447, 485, 511, 629, 655, 685, 697, 771, 831, 879, 959, 965, 1011, 1139, 1241, 1285, 1405, 1535, 1563, 1649, 1731, 1779, 1791, 1799, 1923, 1983, 2005, 2019, 2031, 2043, 2045, 2227, 2605, 2735, 2815, 2827, 2885, 3099

Offset: 1

Author

Robert Israel, Sep 03 2023

Keywords

• nonn
• base

Comments

Odd composite numbers k such that for all divisors d of k, A000120(d) * A000120(k/d) = A000120(k).

Examples

			a(4) = 63 is a term because 63 = 3 * 21 = 7 * 9 with A000120(63) = 6, A000120(3) * A000120(21) = 2 * 3 = 6 and A000120(7) * A000120(9) = 3 * 2 = 6.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..7500

Crossrefs

Cf. A000120, A349494.

Includes x^3 for x in A019434.

Includes all members of A235040 except 1.

Programs

Maple

g:= proc(n) convert(convert(n, base, 2), `+`) end proc:
filter:= proc(n) local d, t;
  if isprime(n) then return false fi;
  t:= g(n);
  andmap(d -> g(d) * g(n/d) = t, select(d -> d^2 <= n, numtheory:-divisors(n)))
end proc:
select(filter, [seq(i,i=3..10000,2)]);

Mathematica

q[n_] := CompositeQ[n] && Ordering[(d = DigitCount[Divisors[n], 2, 1])*Reverse[d], -1][[1]] == Length[d]; Select[Range[3, 3100, 2], q] (* Amiram Eldar, Sep 04 2023 *)

PARI

is(n) = if(n%2 != 1 || isprime(n), return(0)); my(h=hammingweight(n), d=divisors(n), i); for(i=2,(#d+1)\2, if(hammingweight(d[i]) * hammingweight(d[#d+1-i]) > h, return(0))); n > 1 \\ David A. Corneth, Sep 04 2023