A306452 Pseudoprimes to base 3 that are not squarefree, including the non-coprime pseudoprimes.

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

Offset: 1

Author

Jianing Song, Feb 17 2019

Keywords

• nonn

Comments

Numbers k such that 3^k == 3 (mod k) and k is divisible by the square of a Mirimanoff prime (or base-3 Wieferich prime), A014127.

A non-coprime pseudoprime in base b is a number k such that b^k == b (mod k) and that gcd(b, k) > 1, and the non-coprime 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.

Examples

			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
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A013929, A014127, A122780, A158358, A244065, A306450.

Programs

Mathematica

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

PARI

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