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

Author

Amiram Eldar and Thomas Ordowski, Nov 16 2019

Keywords

• nonn

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.

Crossrefs

Cf. A002997, A111076, A329799.

Subsequence of A262043.

Programs

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 *)