A306452
Pseudoprimes to base 3 that are not squarefree, including the non-coprime pseudoprimes.
Original entry on oeis.org
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
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
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Select[Range[5000], PowerMod[3, #, #] == 3 && MoebiusMu[#] == 0 &] (* Alonso del Arte, Mar 16 2019 *)
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forcomposite(n=1, 10^7, if(Mod(3, n)^n==3 && !issquarefree(n), print1(n, ", ")))
A353221
Squarefree base-2 Fermat pseudoprimes divisible by a Wieferich prime.
Original entry on oeis.org
3581761, 5173169, 5968873, 23872213, 36974341, 53711113, 107023281, 427294141, 490950461, 526359289, 546649741, 550230409, 753532781, 803264281, 836683849, 1115578101, 1168492417, 1193557093, 1540454761, 1632785701, 2129304997, 2295209281, 2677147201
Offset: 1
3581761 = 29 * 113 * 1093, so it is a base-2 pseudoprime divisible by the Wieferich prime 1093 and is squarefree.
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/* The following program is valid up to the current search bound for Wieferich primes, about 10^19 as of May 03 2022 (cf. PrimeGrid); the program may miss terms above that bound if there is another Wieferich prime */
forcomposite(c=1, , if(Mod(2, c)^(c-1)==1, if(Mod(c, 1093)==0 || Mod(c, 3511)==0, if(issquarefree(c), print1(c, ", ")))))
A376304
Strong pseudoprimes to base 2 that are not squarefree.
Original entry on oeis.org
1194649, 12327121, 3914864773, 5654273717, 26092328809, 58706246509, 74795779241, 237865367741, 467032496113, 601401837037, 1101047056201, 1629827375177, 2327330361721, 3427506518801, 3950198906473, 6151420925105, 7816904988985, 16034307692677
Offset: 1
a(5) = 26092328809 = 21841 * 1093^2 is a strong pseudoprime that is not squarefree.
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