

A306452


Pseudoprimes to base 3 that are not squarefree, including the noncoprime pseudoprimes.


1



121, 726, 3751, 4961, 7381, 11011, 29161, 32791, 142901, 228811, 239701, 341341, 551881, 566401, 595441, 671671, 784201, 856801, 1016521, 1053426, 1237951, 1335961, 1433971, 1804231, 1916761, 2000251, 2254351, 2446741, 2817001, 2983981, 3078361, 3307051, 3562361
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Numbers k such that 3^k == 3 (mod k) and k is divisible by the square of a Mirimanoff prime (or base3 Wieferich prime), A014127.
A noncoprime pseudoprime in base b is a number k such that b^k == b (mod k) and that gcd(b, k) > 1, and the noncoprime pseudoprime in base 3 (726, 1053426, 6498426, ...) that are not squarefree are listed in A306450 while the others terms in this sequence (121, 3751, 4961, ...) are listed in A244065. So this sequence is the union of A244065 and A306450.
Intersection of A122780 and A013929.


LINKS

Table of n, a(n) for n=1..33.


EXAMPLE

121 is a term because 3^120 == (3^5)^24 == 1 (mod 121) and 121 = 11^2.
Although 3^725 = 243 rather than 1 mod 726, we see that nevertheless 3^726 = 3 mod 726, and since 726 = 2 * 3 * 11^2, 726 is in the sequence.  Alonso del Arte, Mar 16 2019


MATHEMATICA

Select[Range[5000], PowerMod[3, #, #] == 3 && MoebiusMu[#] == 0 &] (* Alonso del Arte, Mar 16 2019 *)


PROG

(PARI) forcomposite(n=1, 10^7, if(Mod(3, n)^n==3 && !issquarefree(n), print1(n, ", ")))


CROSSREFS

Cf. A122780, A158358, A244065, A306450.
Sequence in context: A190877 A293566 A293507 * A238250 A293588 A203959
Adjacent sequences: A306449 A306450 A306451 * A306453 A306454 A306455


KEYWORD

nonn


AUTHOR

Jianing Song, Feb 17 2019


STATUS

approved



