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A383364 a(n) is the least number k with exactly n proper divisors, where all of them have binary weights that are different from the binary weight of k.

%I A383364 #7 Apr 24 2025 12:32:07
%S A383364 1,3,25,15,81,63,15625,231,1225,405,59049,495,531441,5103,2025,1485,
%T A383364 33232930569601,2475,3814697265625,6237,18225,295245,31381059609,4095,
%U A383364 1500625,2657205,81225,25515,22876792454961,14175,931322574615478515625,21735,31236921,301327047
%N A383364 a(n) is the least number k with exactly n proper divisors, where all of them have binary weights that are different from the binary weight of k.
%H A383364 Amiram Eldar, <a href="/A383364/b383364.txt">Table of n, a(n) for n = 0..99</a>
%e A383364 a(0) = 1 since 1 has no proper divisors.
%e A383364 a(1) = 3 since 3 has one proper divisor, 1, and A000120(1) = 1 != A000120(3) = 2, while 2 also has one proper divisor, 1, but A000120(2) = A000120(1) = 1.
%t A383364 q[k_] := DivisorSum[k, 1 &, DigitCount[#, 2, 1] == DigitCount[k, 2, 1] &] == 1; seq[len_] := Module[{s = Table[0, {len}], c = 0, k = 1, i}, While[c < len, i = DivisorSigma[0, k]; If[i <= len && s[[i]] == 0 && q[k], c++; s[[i]] = k]; k++]; s]; seq[16]
%o A383364 (PARI) is1(k) = {my(h = hammingweight(k)); sumdiv(k, d, hammingweight(d) == h) ==  1};
%o A383364 list(len) = {my(s = vector(len), c = 0, k = 1, i); while(c < len, i = numdiv(k); if(i <= len && s[i] == 0 && is1(k), c++; s[i] = k); k++); s;}
%Y A383364 Cf. A000120, A032741, A380844, A383363, A383365.
%K A383364 nonn,base
%O A383364 0,2
%A A383364 _Amiram Eldar_, Apr 24 2025