A339532 Numbers b > 1 such that the smallest three primes, i.e., 2, 3 and 5 are base-b Wieferich primes.

449, 557, 593, 649, 701, 757, 793, 901, 1349, 1457, 1493, 1549, 1601, 1657, 1693, 1801, 2249, 2357, 2393, 2449, 2501, 2557, 2593, 2701, 3149, 3257, 3293, 3349, 3401, 3457, 3493, 3601, 4049, 4157, 4193, 4249, 4301, 4357, 4393, 4501, 4949, 5057, 5093, 5149, 5201

Offset: 1

Author

Felix Fröhlich, Dec 08 2020

Keywords

• nonn

Crossrefs

Cf. A256236. Row 1 of A319060.

Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Programs

Mathematica

Select[Range[2, 5250], Function[b, AllTrue[{2, 3, 5}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)

PARI

is(n) = forprime(p=1, 5, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

Formula

Conjectures from Chai Wah Wu, Aug 18 2025: (Start)

a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.

G.f.: x*(-x^8 + 108*x^7 + 36*x^6 + 56*x^5 + 52*x^4 + 56*x^3 + 36*x^2 + 108*x + 449)/(x^9 - x^8 - x + 1). (End)