A080026 Numbers k having exactly one divisor d such that in binary representation d and k/d have the same number of 1's as k.

1, 9, 49, 225, 961, 3969, 6241, 8281, 16129, 24649, 25281, 33489, 34969, 65025, 82369, 100489, 101761, 123201, 133225, 140625, 143641, 198025, 261121, 328329, 330625, 405769, 408321, 494209, 540225, 564001, 576081, 582169, 664225, 797449, 950625, 1046529, 1147041

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = m^2 with A000120(m) = A000120(n).

A080024(a(n)) = 1.

Example

6241 = 79^2: 1100001100001 = 1001111*1001111, therefore 6241 is a term.

Mathematica

Do[b = Count[ IntegerDigits[n^2, 2], 1]; If[ Count[ IntegerDigits[n, 2], 1] == b, c = 0; d = IntegerDigits[ Divisors[n^2], 2]; l = DivisorSigma[0, n^2]; k = 1; While[ k < Ceiling[l/2], If[Count[d[[k]], 1] == b && Count[d[[l - k + 1]], 1] == b, c++ ]; k++ ]; If[c == 0, Print[n^2]]], {n, 1, 1000}]
dnd1Q[n_]:=Count[Divisors[n], _?(DigitCount[n, 2, 1]==DigitCount[ #, 2, 1] == DigitCount[n/#, 2, 1]&)]==1; Select[Range[800000], dnd1Q] (* Harvey P. Dale, Aug 03 2021 *)

Prog

(PARI) list(lim) = my(h); for(k = 1, lim, h = hammingweight(k); if(hammingweight(k^2) == h && sumdiv(k, d, d < k && hammingweight(d) == h && hammingweight(k^2/d) == h) == 0, print1(k^2, ", "))); \\ Amiram Eldar, Jul 30 2025

Crossrefs

Cf. A000005, A000120, A000196, A000290, A007088, A080024.

Subsequence of A080025.

Sequence in context: A354657 A003297 A012248 * A060867 A192814 A228018

Adjacent sequences: A080023 A080024 A080025 * A080027 A080028 A080029

Keyword

nonn,base

Author

Reinhard Zumkeller, Jan 21 2003

Extensions

More terms from Robert G. Wilson v, Jan 24 2003

Status

approved