

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
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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 n1 (by analogy to the Korselt's criterion).
Gpf(n) is odd, so gpf(n)1 is even. Therefore, n1 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) && (n1) % (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



