A232505 - OEIS

%I #27 May 08 2021 06:58:51

%S 1,1,1,1,18,13,38,50,26,83,66,137,224,242,147,303,509,395,578,364,714,

%T 563,965,1046,1254,1155,1043,1565,1323,1676,1667,2440,2456,2589,2563,

%U 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

%N Sum of odd quadratic residues of prime(n).

%C Seems to have no modular form.

%H Indranil Ghosh, <a href="/A232505/b232505.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Quadratic_residue">Quadratic residue</a>

%F a(n) = A232597(A000040(n)). - _Antti Karttunen_, Nov 26 2013

%e 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.

%e 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.

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

%o (PARI) A232597(n) = {s=0; for(k=1, n, s=s+((k%2)*((1+kronecker(k, n))\2)*k)); return(s); }

%o forprime (i=1, 300, print1(A232597(i), ", ")) \\ _Antti Karttunen_, Nov 26 2013

%o (Python)

%o from sympy.ntheory.residue_ntheory import quadratic_residues as q

%o from sympy import prime

%o def a(n): return sum(i for i in q(prime(n)) if i%2)

%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, May 14 2017

%Y Cf. A076409, A232597.

%K nonn

%O 1,5

%A _Jon Perry_, Nov 25 2013

%E Missing 1 (as a(1) is value for the first prime, 2) inserted into beginning by _Antti Karttunen_, Nov 26 2013