A339531 Numbers b > 1 such that the smallest two primes, i.e., 2 and 3 are base-b Wieferich primes.

17, 37, 53, 73, 89, 109, 125, 145, 161, 181, 197, 217, 233, 253, 269, 289, 305, 325, 341, 361, 377, 397, 413, 433, 449, 469, 485, 505, 521, 541, 557, 577, 593, 613, 629, 649, 665, 685, 701, 721, 737, 757, 773, 793, 809, 829, 845, 865, 881, 901, 917, 937, 953

Offset: 1

Author

Felix Fröhlich, Dec 08 2020

Keywords

• nonn

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A256236, A263941. Row 1 of A319059.

Cf. smallest k primes are base-b Wieferich primes: A339532 (k=3), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Programs

Mathematica

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

PARI

is(n) = forprime(p=1, 3, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

Formula

a(n) = 4*A263941(n) + 1 for n>=2, a(n) = 4*floor((9*n)/2) + 1 for all n. - Hugo Pfoertner, Dec 08 2020

From Chai Wah Wu, Aug 18 2025: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.

G.f.: x*(-x^2 + 20*x + 17)/((x - 1)^2*(x + 1)). (End)