A002143 -id:A002143 - cp's OEIS frontend

A125616 (Sum of the quadratic nonresidues of prime(n)) / prime(n).

1, 2, 3, 3, 4, 5, 7, 7, 9, 9, 10, 11, 14, 13, 16, 15, 17, 21, 18, 22, 22, 22, 24, 25, 28, 28, 27, 28, 34, 35, 34, 36, 37, 41, 39, 41, 47, 43, 47, 45, 54, 48, 49, 54, 54, 59, 59, 57, 58, 67, 60, 66, 64, 72, 67, 73, 69, 70, 72, 73, 78, 87, 78, 79, 84, 84, 89, 87, 88, 99, 96, 93, 96

Offset: 3

Nick Hobson, Nov 30 2006

Always an integer for primes >= 5.

			The quadratic nonresidues of 7=prime(4) are 3, 5 and 6. Hence a(4) = (3+5+6)/7 = 2.

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 185.

Nick Hobson, Table of n, a(n) for n = 3..1000

Cf. A002143, A076409, A076410, A125613-A125618.

a:= proc(n) local p;
   p:= ithprime(n);
   convert(select(t->numtheory:-legendre(t,p)=-1, [$1..p-1]),`+`)/p;
end proc:
seq(a(n),n=3..100); # Robert Israel, May 10 2015

Table[Total[Flatten[Position[Table[JacobiSymbol[a, p], {a, p - 1}], -1]]]/ p, {p, Prime[Range[3, 100]]}] (* Geoffrey Critzer, May 10 2015 *)

vector(73, m, p=prime(m+2); t=1; for(i=2, (p-1)/2, t+=((i^2)%p)); (p-1)/2-t/p)

a(n) = A125615(n)/prime(n).

If prime(n) = 4k+1 then a(n) = k = A076410(n).

A355879 Class number of Q(sqrt((-1)^((p-1)/2)*p)), where p = prime(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 7, 1, 5, 3, 1, 1, 1, 5, 3, 1, 1, 5, 5, 1, 3, 1, 7, 1, 1, 11, 1, 5, 1, 13, 1, 1, 9, 3, 7, 5, 3, 1, 15, 1, 7, 3, 13, 1, 11, 1, 1, 3, 1, 3, 19, 1, 1, 3, 1, 5, 1, 1, 19, 9, 1, 3, 17, 1, 1, 5, 1, 9, 1, 21, 1, 15, 5, 1, 1, 1, 7

Offset: 1

Jianing Song, Jul 20 2022

nonn

For n > 1, class number of the unique quadratic field with discriminant +-p, p = prime(n).
a(1) corresponds to Q(sqrt(2*i)) = Q(1+i) = Q(i).
All terms are odd.

			prime(9) = 23, Q(sqrt(-23)) has class number 3, so a(9) = 3.
prime(15) = 47, Q(sqrt(-47)) has class number 5, so a(15) = 5.
prime(20) = 71, Q(sqrt(-71)) has class number 7, so a(20) = 7.
prime(50) = 229, Q(sqrt(229)) has class number 3, so a(50) = 3.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Class numbers of quadratic fields
Index entries for sequences related to quadratic fields

Cf. A002143, A002146.

a(n) = if(n==1, 1, my(p=prime(n)); qfbclassno(if(p%4==1, p, -p)))

A101435 Dimension of a certain space of modular forms of weight 2 and level p^2, where p runs through the primes > 3 that are == 3 mod 4. See reference for precise definition.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 11, 11, 11, 13, 13, 15, 15, 17, 17, 17, 19, 19, 21, 21, 23, 23, 23, 25, 27, 27, 29, 31, 31, 31, 33, 35, 37, 37, 37, 39, 39, 41, 41, 41, 41, 43, 43, 45, 47, 47, 49, 51, 51, 51, 53, 53, 55, 55, 57, 57, 61, 61, 61, 63, 63, 65, 67, 69, 69, 71, 71, 73

Offset: 1

N. J. A. Sloane, Nov 29 2006

nonn

A. Pacetti and F. Rodriguez Villegas, Computing weight two modular forms of level p^2, Math. Comp. 74 (2004), 1545-1557. See Table 1.

Cf. A002143.

with(numtheory); L:=legendre; f:=p->(p+5)/12 + (1-L(-3,p))/3-(1-L(2,p))/2;

A165186 a(n) = Sum_{k=1..n} (k*(n-k) mod n).

Original entry on oeis.org

0, 1, 4, 6, 10, 17, 28, 36, 30, 45, 66, 82, 78, 105, 140, 136, 136, 141, 190, 230, 238, 253, 322, 380, 250, 325, 360, 434, 406, 505, 558, 592, 572, 561, 700, 678, 666, 741, 910, 980, 820, 917, 946, 1122, 1050, 1173, 1316, 1432, 1078, 1125, 1394, 1430, 1378, 1449

Offset: 1

Wouter Meeussen, Sep 06 2009

nonn

Comment from Max Alekseyev, Nov 22 2009: For a prime p==3 (mod 4), a(p) = p*h(-p) + p*(p-1)/2 where h(-p) is the class number (listed in A002143). For example, h(-19)=1 and a(19) = 19*1 + 19*18/2 = 190.

Cf. A008784, A048153

Table[Sum[Mod[k (n-k),n],{k,n}],{n,100}]

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A125616 (Sum of the quadratic nonresidues of prime(n)) / prime(n).

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A355879 Class number of Q(sqrt((-1)^((p-1)/2)*p)), where p = prime(n).

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A101435 Dimension of a certain space of modular forms of weight 2 and level p^2, where p runs through the primes > 3 that are == 3 mod 4. See reference for precise definition.

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Maple

A165186 a(n) = Sum_{k=1..n} (k*(n-k) mod n).

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Mathematica