A307434 a(n) is the smallest k such that the p-rank of (Z/kZ)* is 2, where p = prime(n) and (Z/kZ)* is the multiplicative group of integers modulo n.
8, 63, 275, 1247, 1541, 4187, 14111, 43739, 6533, 13747, 116003, 33227, 61337, 74563, 186497, 79501, 586343, 269011, 432821, 485357, 128627, 451091, 83333, 191351, 377719, 491063, 638189, 551051, 2617309, 359341, 1164083, 206981, 902831, 929633, 2134277
Offset: 1
Keywords
Examples
(Z/8Z)* = C_2 X C_2, in which the solutions to x^2 == 1 (mod 8) are x == 1, 3, 5, 7 (mod 8) (4 solutions); (Z/63Z)* = C_6 X C_6, in which the solutions to x^3 == 1 (mod 63) are x == 1, 4, 16, 22, 25, 37, 43, 46, 58 (mod 63) (9 solutions); (Z/275Z)* = C_10 X C_20, in which the solutions to x^5 == 1 (mod 275) are x == 1, 16, 26, 31, 36, 56, 71, 81, 86, 91, 111, 126, 136, 141, 146, 166, 181, 191, 196, 201, 221, 236, 246, 251, 256 (mod 275) (25 solutions).
Links
- Jianing Song, Table of n, a(n) for n = 1..10000
- Jianing Song, Factorization of a(n) for n = 4..10000
- Wikipedia, Multiplicative group of integers modulo n
Programs
-
PARI
a(n) = if(n==1, 8, my(p=prime(n), i=0, q=0); for(k=1, +oo, if(isprime(2*k*p+1), i++; if(i==1, q=2*k*p+1)); if(i==2, return(q*min(p^2, 2*k*p+1)))))
Formula
a(1) = 8; a(n) = A307436(prime(n)) for n >= 2.
Comments