A329538 Odd composite numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).

29341, 1152271, 5481451, 14913991, 15247621, 36765901, 133800661, 178482151, 299736181, 579606301, 652969351, 702683101, 739444021, 743404663, 775368901, 3215031751, 4340265931, 5871134179, 8657319259, 9293756581, 12191597551, 13734086221, 14386156093, 19331388805

Offset

1,1

Comments

Carmichael numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).

Note that if p is an odd prime, then A111076(p)^((p-1)/2) == -1 (mod p).

Max Alekseyev proved (in a letter to the second author) that all these numbers have an odd number of prime factors, showing that if k is a term, then k is a Carmichael number m such that p-1 does not divide (m-1)/2 for every prime p|m (the numbers m form the supersequence A329799).

There are 6469 terms k of this sequence below 2^64:

4240 with 3 prime factors, least is 29341 = 13*37*61,

1790 with 5 prime factors, least is 4340265931 = 19*43*107*131*379,

437 with 7 prime factors, least is 37038179683765 = 5*13*29*37*317*757*2213,

2 with 9 prime factors, least is 1025735495681200591 = 7*19*31*67*79*163*199*271*5347.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Index entries for sequences related to Carmichael numbers.

Mathematica

f[1, lam_] = 1; f[n_, lam_] := If[n < 5, n - 1, Module[{k = 1}, While[GCD[k, n] > 1 || MultiplicativeOrder[k, n] < lam, k++]; k]]; aQ[n_] := CompositeQ[n] && Divisible[n - 1 , (lam = CarmichaelLambda[n])] && PowerMod[f[n, lam], (n - 1)/2, n] == n - 1; Select[Range[1, 6*10^6, 2], aQ] (* after the Charles R Greathouse IV at A111076 *)

Crossrefs

Cf. A002997, A111076, A329799.

Subsequence of A262043.

Sequence in context: A186566 A083740 A083739 * A182133 A182416 A232201

Adjacent sequences: A329535 A329536 A329537 * A329539 A329540 A329541

Keyword

nonn

Author

Amiram Eldar and Thomas Ordowski, Nov 16 2019

Status

approved