A375875 Smallest n-bit Carmichael number, or 0 if no such number exists.

561, 1105, 2465, 6601, 8911, 29341, 41041, 75361, 162401, 278545, 530881, 1050985, 2100901, 4335241, 8719309, 16778881, 33596641, 67371265, 134809921, 270857521, 540066241, 1074363265, 2159003281, 4295605861, 8612234401, 17190510961, 34364331001, 68910004801, 137691502081, 274895715601, 549813672001, 1099910311201

Offset

10,1

Comments

a(n) > 0 for large enough n, see Larsen link. Probably a(n) > 0 for all n >= 10.

Links

Amiram Eldar, Table of n, a(n) for n = 10..74

Daniel Larsen, Bertrand's Postulate for Carmichael Numbers, International Mathematics Research Notices, Vol. 2023, No. 15 (2023), pp. 13072-13098; arXiv preprint, arXiv:2111.06963 [math.NT], 2021-2023.

Formula

a(n) ~ 2^n due to Larsen.

Mathematica

carmQ[k_] := CompositeQ[k] && Divisible[k-1, CarmichaelLambda[k]]; a[n_] := Module[{k = 2^(n-1)}, While[!carmQ[k], k++]; If[k > 2^n, 0, k]]; Array[a, 10, 10] (* Amiram Eldar, Feb 04 2026 *)

Prog

(PARI) a(n)=forsquarefree(t=2^(n-1), 2^n, my(f=t[2]); if(#f~>1 && f[1, 1]>2 && Korselt(t[1], f), return(t[1])))

Crossrefs

Conjecturally a subsequence of A002997.

Sequence in context: A263403 A083733 A339869 * A214428 A262043 A264012

Adjacent sequences: A375872 A375873 A375874 * A375876 A375877 A375878

Keyword

nonn

Author

Charles R Greathouse IV, Sep 01 2024

Status

approved