A191311 Numbers n such that exactly half of the a such that 0

4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, 23001, 30889, 38503, 39865, 49141, 68101, 79003, 88561, 88831, 91001, 93961, 104653, 107185, 137149, 146611, 152551, 157641, 176149, 188191, 204001, 218791, 226801, 228241

Offset: 1

Author

Jason Holt, Jun 04 2011

Keywords

• easy
• nonn

Comments

Values of n for which half the witnesses in the Fermat primality test are false.

When n=pq with p,q=2p-1 prime, a^(n-1) = 1 (mod p) iff a is a quadratic residue mod q. So A129521 is a subsequence. - Gareth McCaughan, Jun 05 2011

From Robert G. Wilson v, Aug 13 2011: (Start)

Number of terms less than 10^n: 2, 4, 5, 7, 22, 60, 129, 303, 690, 1785, …, .

In reference to the numbers in the b-file: (1) number of terms which have k>0 prime factors: 1, 1058, 139, 512, 339, 102, 6; (2) about half of the terms, 1058, are members of A129521, those which have just two prime factors; (3) except for the first term, all terms are squarefree, except for the first two terms, all terms are odd; and (4) most terms, more than 98.5%, are congruent to 1 modulo 6. (End)

Links

• David W. Wilson and Robert G. Wilson v, Table of n, a(n) for n = 1..2157
• While exploring Carmichael numbers, I noticed a few values on the chart on this page for which exactly half of the relatively prime witnesses to the Fermat primality test were false witnesses.

Crossrefs

A063994 gives the number of false witnesses for every n.

A129521 is a subsequence. See also A191592.

Programs

Mathematica

fQ[n_] := Block[{pf = First /@ FactorInteger@ n}, 2Times @@ GCD[n - 1, pf - 1] == n*Times @@ (1 - 1/pf)]; Select[ Range@ 250000, fQ] (* Robert G. Wilson v, Aug 08 2011 *)

Python

import math
for x in range(2, 1000):
  false_witnesses = 0
  relatively_prime_values = 0
  for y in range(x):
    if math.gcd(y, x) == 1:
      relatively_prime_values += 1
    if (pow(y, x-1, x) == 1):
      false_witnesses += 1
  if false_witnesses * 2 == relatively_prime_values:
    print(x, "is a Fermat Half-Prime")

Formula

Integers, n, such that A063994(n) = 2*A000010(n). - Robert G. Wilson v, Aug 13 2011

Extensions

Edited by N. J. A. Sloane, Jun 07 2011. I made use of a more explicit definition due to Gareth McCaughan, Jun 05 2011.