A242736 Number of solutions of a^2 + b^2 congruent to -1 modulo the n-th prime.

0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 13, 13, 13, 14, 14, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 25, 27, 28, 29, 29, 30, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 39, 39, 40, 40, 42, 43, 44, 44

Offset

1,5

Comments

a(n) is the number of solutions of a^2 + b^2 congruent to -1 modulo the n-th prime, where 0 <= a <= b <= floor((p-1)/2).

Is this sequence nondecreasing? The data for the first thousand terms supports this conjecture.

Links

Table of n, a(n) for n=1..70.

MathOverflow, How does this sequence grow

Formula

See the answer given in the link above.

For n>1, a(n) = ceiling(p(n)/8), where p(n) is the n-th prime.

Example

For n=5: The 5th prime is 11. 1^2 + 3^2 = 10 is congruent to -1 (mod 11) and 4^2 + 4^2 = 32 is congruent to -1 (mod 11).

Maple

A242736:=n->ceil(ithprime(n)/8): 0, seq(A242736(n), n=2..100); # Wesley Ivan Hurt, Dec 12 2015

Mathematica

iend=50;
For[n=1, n<=iend, n++,
p=Prime[n];
count[n]=0;
For[i=0, i<=(p-1)/2, i++,
For[j=i, j<=(p-1)/2, j++,
If[Mod[i^2+j^2, p]==p-1, count[n]++; ]]]]
Print[Table[count[i], {i, 1, iend}]]

Crossrefs

Sequence in context: A025777 A269862 A194200 * A194237 A145707 A145703

Adjacent sequences: A242733 A242734 A242735 * A242737 A242738 A242739

Keyword

nonn,easy

Author

David S. Newman, May 21 2014

Extensions

More terms from Alois P. Heinz, Jun 17 2014

Status

approved