A270819 a(n) is the number of arithmetic progressions of length 3 among the quadratic residues modulo prime(n).

0, 0, 0, 0, 10, 12, 16, 36, 44, 84, 90, 144, 160, 210, 230, 312, 406, 420, 528, 560, 576, 702, 820, 880, 1056, 1200, 1224, 1378, 1404, 1456, 1890, 2080, 2176, 2346, 2664, 2700, 2964, 3240, 3320, 3612, 3916, 3960, 4370, 4416, 4704, 4752, 5460, 5994, 6328, 6384, 6496

Offset: 1

Author

Michel Marcus, Mar 23 2016

Keywords

• nonn

Comments

Wraparound progressions as well as decreasing progressions are counted.

Examples

			For p=prime(5)=11, whose quadratic residues are (1,3,4,5,9), some examples of 3-term arithmetic progressions are (3,4,5), (4,9,3) and (5,4,3).
		

References

• R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.29 p. 111.

Crossrefs

Cf. A063987.

Programs

Mathematica

Table[(# - 1) Floor[(# - 2)/8] &@ Prime@ n, {n, 51}] (* Michael De Vlieger, Mar 23 2016 *)

PARI

a(n) = my(p=prime(n)); (p-1)*((p-2)\8);

Formula

a(n) = (prime(n)-1)*floor((prime(n)-2)/8).