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)
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
KEYWORD
nonn
AUTHOR
Amiram Eldar and Thomas Ordowski, Nov 16 2019
STATUS
approved