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

%I A365451 #16 Sep 04 2023 12:26:21
%S A365451 15,27,51,63,85,95,111,119,123,125,187,219,221,255,335,365,411,447,
%T A365451 485,511,629,655,685,697,771,831,879,959,965,1011,1139,1241,1285,1405,
%U A365451 1535,1563,1649,1731,1779,1791,1799,1923,1983,2005,2019,2031,2043,2045,2227,2605,2735,2815,2827,2885,3099
%N A365451 Odd composite numbers k such that A349494(k) = A000120(k).
%C A365451 Odd composite numbers k such that for all divisors d of k, A000120(d) * A000120(k/d) = A000120(k).
%H A365451 Robert Israel, <a href="/A365451/b365451.txt">Table of n, a(n) for n = 1..7500</a>
%e A365451 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.
%p A365451 g:= proc(n) convert(convert(n, base, 2), `+`) end proc:
%p A365451 filter:= proc(n) local d, t;
%p A365451   if isprime(n) then return false fi;
%p A365451   t:= g(n);
%p A365451   andmap(d -> g(d) * g(n/d) = t, select(d -> d^2 <= n, numtheory:-divisors(n)))
%p A365451 end proc:
%p A365451 select(filter, [seq(i,i=3..10000,2)]);
%t A365451 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 *)
%o A365451 (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
%Y A365451 Cf. A000120, A349494.
%Y A365451 Includes x^3 for x in A019434.
%Y A365451 Includes all members of A235040 except 1.
%K A365451 nonn,base
%O A365451 1,1
%A A365451 _Robert Israel_, Sep 03 2023