A383365 Numbers k with a record number of proper divisors, where all of them have binary weights that are different from the binary weight of k.

1, 3, 15, 63, 231, 405, 495, 1485, 2475, 4095, 14175, 21735, 24255, 31185, 79695, 190575, 218295, 239085, 294525, 904365, 1276275, 2789325, 3586275, 4937625, 6912675, 10072755, 17342325, 17972955, 26801775, 46621575, 80405325, 192567375, 326351025, 333107775, 654729075

Offset

1,2

Comments

The corresponding record values are 0, 1, 3, 5, 7, 9, 11, 15, 17, ... (see the link for more values).

All terms are odd as an even number k has proper divisor k/2 with the same binary weight. - David A. Corneth, Apr 24 2025

Links

David A. Corneth, Table of n, a(n) for n = 1..91 (first 46 terms from Amiram Eldar)

Amiram Eldar, Table of n, a(n), A032741(a(n)) for n = 1..46.

Example

a(1) = 1 since 1 has no proper divisors.
a(2) = 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.
a(3) = 15 since 15 has 3 proper divisors, 1, 3 and 5, and A000120(1) = 1 and A000120(3) = A000120(5) = 2 are different from A000120(15) = 4. All the numbers below 15 have fewer proper divisors with this property.

Mathematica

q[k_] := DivisorSum[k, 1 &, DigitCount[#, 2, 1] == DigitCount[k, 2, 1] &] == 1; seq[kmax_] := Module[{s = {}, d, dm = 0}, Do[d = DivisorSigma[0, k]; If[d > dm && q[k], dm = d; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^5]

Prog

(PARI) is1(k) = {my(h = hammingweight(k)); sumdiv(k, d, hammingweight(d) == h) ==  1};
list(kmax) = {my(d, dm = 0); for(k = 1, kmax, d = numdiv(k); if(d > dm && is1(k), dm = d; print1(k, ", "))); }

Crossrefs

Cf. A000120, A032741, A380844, A383363, A383364.

Sequence in context: A204086 A379242 A171761 * A229277 A218313 A218190

Adjacent sequences: A383362 A383363 A383364 * A383366 A383367 A383368

Keyword

nonn,base

Author

Amiram Eldar, Apr 24 2025

Status

approved