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A276832 Squarefree composite numbers n such that b^n == b (mod gpf(n)) for every integer b, where gpf(n) = A006530(n). 2
231, 561, 1045, 1105, 1653, 1729, 2465, 2821, 3059, 3655, 4371, 4641, 5005, 5083, 5365, 5565, 6545, 6601, 7337, 8029, 8695, 8911, 9361, 10011, 10585, 10857, 10879, 11077, 11305, 12121, 13685, 13695, 15181, 15753, 15841, 16269, 17755, 18361, 18565, 19721, 20301, 20501, 21115 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It suffices to check all bases 2 <= b <= (gpf(n)+1)/2.

Squarefree composite numbers n such that gpf(n)-1 divides n-1 (by analogy to the Korselt's criterion).

Gpf(n) is odd, so gpf(n)-1 is even. Therefore, n-1 is even so n is odd. - David A. Corneth, Sep 20 2016

These numbers have at least three prime factors. Carmichael numbers A002997 are a subsequence. So the sequence is infinite. - Thomas Ordowski and Altug Alkan, Sep 20 2016

First numbers with 3, 4, ... prime factors are 231 = 3 * 7 * 11, 4641 = 3 * 7 * 13 * 17, 31395 = 3 * 5 * 7 * 13 * 23, 1163085 = 3 * 5 * 7 * 11 * 19 * 53, 11996985 = 3 * 5 * 7 * 11 * 13 * 17 * 47, and 286140855 = 3 * 5 * 7 * 11 * 13 * 17 * 19 * 59. - Charles R Greathouse IV, Sep 20 2016

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

231 = 3*7*11 is a term because b^231 == b (mod 11) for every integer b.

45 = 3^2*5 is not a term because 45 is not squarefree although b^45 == b (mod 5) for every integer b.

MATHEMATICA

Select[Select[DeleteCases[Range[2, 22000], p_ /; PrimeQ@ p], SquareFreeQ], Divisible[# - 1, FactorInteger[#][[-1, 1]] - 1] &] (* Michael De Vlieger, Sep 20 2016 *)

PROG

(PARI) A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);

is(n) = n>1 && issquarefree(n) && (n-1) % (A006530(n)-1) == 0 && !isprime(n)

CROSSREFS

Cf. A002997, A006530, A276818.

Sequence in context: A160355 A211712 A324315 * A324319 A246886 A258167

Adjacent sequences:  A276829 A276830 A276831 * A276833 A276834 A276835

KEYWORD

nonn

AUTHOR

Thomas Ordowski and Altug Alkan, Sep 20 2016

STATUS

approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)