A000790 Primary pretenders: least composite c such that n^c == n (mod c).

4, 4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, 4, 4, 6, 6, 4, 4, 6, 9, 4, 4, 38, 6, 4, 4, 6, 6, 4, 4, 6, 46, 4, 4, 10, 6, 4, 4, 6, 6, 4, 4, 6, 15, 4, 4, 9, 6, 4, 4, 6, 6, 4, 4, 6, 9, 4, 4, 15, 6, 4, 4, 6, 6, 4, 4, 6, 21, 4, 4, 10, 6, 4

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

It is remarkable that this sequence is periodic with period 19568584333460072587245340037736278982017213829337604336734362\ 294738647777395483196097971852999259921329236506842360439300 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37 * 41 * 43 * 47 * 53 * 59 * 61 * 67 * 71 * 73 * 79 * 83 * 89 * 97 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 * 139 * 149 * 151 * 157 * 163 * 167 * 173 * 179 * 181 * 191 * 193 * 197 * 199 * 211 * 223 * 227 * 229 * 233 * 239 * 241 * 251 * 257 * 263 * 269 * 271 * 277.

Note that the period is 277# * 23# (where as usual # is the primorial). - Charles R Greathouse IV, Feb 23 2014

Records are 4, 341, 382 & 561, and they occur at indices of 0, 2, 383 & 10103. - Robert G. Wilson v, Feb 22 2014

Andrzej Schinzel (1961) proved that a(n) > 6 if and only if n == {2, 11} (mod 12). - Thomas Ordowski and Krzysztof Ziemak, Jan 21 2018

We have a(n) <= A090086(n), with equality iff gcd(a(n),n) = 1. - Thomas Ordowski, Feb 13 2018

Sequence b(n) = gcd(a(n), n) is also periodic with period P = 23# * 277#, because this is the LCM of all terms, cf. A108574. - M. F. Hasler, Feb 16 2018

Examples

			a(2) = 341 because 2^341 == 2 (mod 341) and there is no smaller composite number c such that 2^c == 2 (mod c).
a(3) = 6 because 3^6 == 3 (mod 6) (whereas 3^4 == 1 (mod 4)).
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• John H. Conway, Richard K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313.
• Steven Finch, The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
• A. Rotkiewicz, Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function lx^C, Acta Arith. XCI.1 (1999), 75-83.
• A. Schinzel, Sur les nombres composés n qui divisent a^n - a, Rend. Circ. Mat. Palermo (2) 7 (1958), 37-41.
• W. Sierpiński, A remark on composite numbers m which are factors of a^m - a, Wiadom. Mat. 4 (1961), 183-184 (in Polish; MR 23#A87).
• OEIS Index to periodic sequences.

Crossrefs

Cf. A108574 (all values occurring in this sequence).

Cf. A002808, A090086, A295997 (it has the same set of distinct terms).

Programs

Haskell

import Math.NumberTheory.Moduli (powerMod)
a000790 n = head [c | c <- a002808_list, powerMod n c c == mod n c]
-- Reinhard Zumkeller, Jul 11 2014

Maple

f:= proc(n) local c;
  for c from 4 do
    if not isprime(c) and n &^ c - n mod c = 0 then return c fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, Jan 21 2018

Mathematica

a[n_] := For[c = 4, True, c = If[PrimeQ[c + 1], c + 2, c + 1], If[PowerMod[n, c, c] == Mod[n, c], Return[c]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 18 2013 *)

PARI

a(n)=forcomposite(c=4,554,if(Mod(n,c)^c==n,return(c))); 561 \\ Charles R Greathouse IV, Feb 23 2014

Python

from sympy import isprime
def A000790(n):
    c = 4
    while pow(n,c,c) != (n % c) or isprime(c):
        c += 1
    return c # Chai Wah Wu, Apr 02 2021