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

1,2

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?

Links

Table of n, a(n) for n=1..58.

Example

(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,

Mathematica

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

Prog

(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, ", ")))))

Crossrefs

Cf. A008330, A055578, A089451, A143548.

Sequence in context: A303716 A309629 A075573 * A302677 A304267 A166138

Adjacent sequences: A317703 A317704 A317705 * A317707 A317708 A317709

Keyword

nonn,tabf

Author

Jianing Song, Aug 05 2018

Status

approved