A232505 Sum of odd quadratic residues of prime(n).

1, 1, 1, 1, 18, 13, 38, 50, 26, 83, 66, 137, 224, 242, 147, 303, 509, 395, 578, 364, 714, 563, 965, 1046, 1254, 1155, 1043, 1565, 1323, 1676, 1667, 2440, 2456, 2589, 2563, 2284, 2827, 3362, 2526, 3503, 4408, 3765, 3271, 4902, 4557, 4005, 5829, 5380, 6952, 6093, 7046, 5288, 7626, 8691, 8552, 6871, 8563, 7622, 9007, 10250, 10365, 10233

Offset: 1

Author

Jon Perry, Nov 25 2013

Keywords

• nonn

Comments

Seems to have no modular form.

Examples

			a(1), a(2), a(3) and a(4) are all 1, as for the corresponding primes 2, 3, 5 and 7 the quadratic residue sets are {1}, {1}, {1,4} and {1,2,4}, in which all cases, only 1 is an odd residue.
For a(5), which is computed for the 5th prime, 11, we have a set of its quadratic residues (those less than 11) as {1,3,4,5,9}, of which when we sum only the odd residues, 1+3+5+9, we get a(5) = 18.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..1000
• Wikipedia, Quadratic residue

Crossrefs

Cf. A076409, A232597.

Programs

Mathematica

Table[Function[p, Total@ Select[Range[1, p, 2], JacobiSymbol[#, p] == 1 &]]@ Prime@ n, {n, 62}] (* Michael De Vlieger, May 14 2017 *)

PARI

A232597(n) = {s=0; for(k=1, n, s=s+((k%2)*((1+kronecker(k, n))\2)*k)); return(s); }
forprime (i=1, 300, print1(A232597(i), ", ")) \\ Antti Karttunen, Nov 26 2013

Python

from sympy.ntheory.residue_ntheory import quadratic_residues as q
from sympy import prime
def a(n): return sum(i for i in q(prime(n)) if i%2)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

Formula

a(n) = A232597(A000040(n)). - Antti Karttunen, Nov 26 2013

Extensions

Missing 1 (as a(1) is value for the first prime, 2) inserted into beginning by Antti Karttunen, Nov 26 2013