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

1,5

Comments

Seems to have no modular form.

Links

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Wikipedia, Quadratic residue

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A076409, A232597.

Sequence in context: A030239 A040308 A077810 * A297939 A298550 A160901

Adjacent sequences: A232502 A232503 A232504 * A232506 A232507 A232508

Keyword

nonn

Author

Jon Perry, Nov 25 2013

Extensions

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

Status

approved