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

1,5

Comments

Wraparound progressions as well as decreasing progressions are counted.

References

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

Links

Table of n, a(n) for n=1..51.

Formula

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

Example

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).

Mathematica

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

Prog

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

Crossrefs

Cf. A063987.

Sequence in context: A109604 A187710 A063096 * A031183 A265043 A158871

Adjacent sequences: A270816 A270817 A270818 * A270820 A270821 A270822

Keyword

nonn

Author

Michel Marcus, Mar 23 2016

Status

approved