A317706 Irregular triangle of numbers k < p^2 such that k is a primitive root modulo p but not p^2, p = prime(n).

1, 8, 7, 18, 19, 31, 40, 94, 112, 118, 19, 80, 89, 150, 40, 65, 75, 131, 158, 214, 224, 249, 116, 127, 262, 299, 307, 333, 28, 42, 63, 130, 195, 263, 274, 352, 359, 411, 14, 60, 137, 221, 374, 416, 425, 467, 620, 704, 781, 827, 115, 117, 145, 229, 414, 513, 623, 726

Offset: 1

Author

Jianing Song, Aug 05 2018

Keywords

• nonn
• tabf

Comments

Also row n lists numbers k < p^2 such that the multiplicative order of k modulo p^2 is p - 1.

Row n has phi(prime(n) - 1) = A008330(n) terms.

Row sum is congruent to mu(prime(n) - 1) = A089451(n) modulo prime(n)^2, where mu is the Moebius function. For n >= 3, the product of n-th row is congruent to 1 modulo prime(n)^2.

Does every integer appear in this sequence? For example, 3 does not appear until the prime 1006003 and 5 does not appear until the prime 40487. Where does 2 first appear?

Examples

			(2)   1,
(3)   8,
(5)   7, 18,
(7)   19, 31,
(11)  40, 94, 112, 118,
(13)  19, 80, 89, 150,
(17)  40, 65, 75, 131, 158, 214, 224, 249,
(19)  116, 127, 262, 299, 307, 333,
(23)  28, 42, 63, 130, 195, 263, 274, 352, 359, 411,
		

Crossrefs

Cf. A008330, A055578, A089451, A143548.

Programs

Mathematica

Table[Select[Range[p^2 - 1], MultiplicativeOrder[#, p^2] == p - 1 &], {p, Prime@ Range@ 11}] // Flatten (* Michael De Vlieger, Aug 05 2018 *)

PARI

forprime(p=2,100,for(i=1,p^2,if(Mod(i,p)!=0,if(znorder(Mod(i,p^2))==p-1,print1(i, ", ")))))