A365451 - OEIS

%I #16 Sep 04 2023 12:26:21

%S 15,27,51,63,85,95,111,119,123,125,187,219,221,255,335,365,411,447,

%T 485,511,629,655,685,697,771,831,879,959,965,1011,1139,1241,1285,1405,

%U 1535,1563,1649,1731,1779,1791,1799,1923,1983,2005,2019,2031,2043,2045,2227,2605,2735,2815,2827,2885,3099

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

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

%H Robert Israel, <a href="/A365451/b365451.txt">Table of n, a(n) for n = 1..7500</a>

%e 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 g:= proc(n) convert(convert(n, base, 2), `+`) end proc:

%p filter:= proc(n) local d, t;

%p   if isprime(n) then return false fi;

%p   t:= g(n);

%p   andmap(d -> g(d) * g(n/d) = t, select(d -> d^2 <= n, numtheory:-divisors(n)))

%p end proc:

%p select(filter, [seq(i,i=3..10000,2)]);

%t 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 (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 Cf. A000120, A349494.

%Y Includes x^3 for x in A019434.

%Y Includes all members of A235040 except 1.

%K nonn,base

%O 1,1

%A _Robert Israel_, Sep 03 2023