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.

1, 3, 25, 15, 81, 63, 15625, 231, 1225, 405, 59049, 495, 531441, 5103, 2025, 1485, 33232930569601, 2475, 3814697265625, 6237, 18225, 295245, 31381059609, 4095, 1500625, 2657205, 81225, 25515, 22876792454961, 14175, 931322574615478515625, 21735, 31236921, 301327047

Offset

0,2

Links

Amiram Eldar, Table of n, a(n) for n = 0..99

Example

a(0) = 1 since 1 has no proper divisors.
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.

Mathematica

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]

Prog

(PARI) is1(k) = {my(h = hammingweight(k)); sumdiv(k, d, hammingweight(d) == h) ==  1};
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; }

Crossrefs

Cf. A000120, A032741, A380844, A383363, A383365.

Sequence in context: A085836 A073916 A076962 * A370416 A304480 A129599

Adjacent sequences: A383361 A383362 A383363 * A383365 A383366 A383367

Keyword

nonn,base

Author

Amiram Eldar, Apr 24 2025

Status

approved