A389280 a(n) is the least number k > 0 such that wt(k^n) >= wt(k) * T(n), or -1 if such k does not exist, where wt(x) = A000120(x) is the binary weight of x, and T(n) = n*(n+1)/2 = A000217(n) is the n-th triangular number.

1, 1, 2697, 645, 266245, 2113793, 2097185, 134217857, 4429185041, 17179869313, 68721573889, 68719477249, 8796093022465, 17592186048513, 562949953421825, 1125902054326273, 288230376151712257, 1152921504606851073, 4611686018427396097, 36893488147419104257

Offset

0,3

Comments

Conjecture: a(n) > 0.

Links

Kevin Ryde, Table of n, a(n) for n = 0..75

Kevin Ryde, C Code

Example

For n=2, 2697 is the least integer k such that wt(k^2) >= wt(k) * 3, therefore a(2)=2697.
For n=4, 266245 is the least integer k such that wt(k^4) >= wt(k) * 10, therefore a(4)=266245.

Prog

(PARI) a(n) = my(k=1, t=n*(n+1)/2); while (hammingweight(k^n) < hammingweight(k)*t, k++); k; \\ Michel Marcus, Oct 03 2025
(Python)
from itertools import count
def A389280(n): return next(k for k in count(1) if (k**n).bit_count()>=k.bit_count()*(n*(n+1)>>1)) # Chai Wah Wu, Oct 10 2025
(C) /* See links. */

Crossrefs

Cf. A000120, A000217, A358268.

Sequence in context: A254554 A254547 A254477 * A115930 A109994 A322525

Adjacent sequences: A389277 A389278 A389279 * A389281 A389282 A389283

Keyword

nonn,base,hard

Author

Alex Ratushnyak, Sep 28 2025

Extensions

a(12) from Chai Wah Wu, Oct 21 2025

a(13) onward from Kevin Ryde, Oct 24 2025

Status

approved