A260928 Number of positive integers k < prime(n)/2 such that k + k' is a square, where k' is the unique integer among 1, ..., prime(n)-1 such that k*k' == 1 (mod prime(n)).

0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 4, 1, 1, 0, 2, 0, 3, 1, 1, 1, 3, 0, 2, 1, 1, 2, 1, 2, 3, 5, 4, 1, 10, 5, 10, 2, 8, 3, 1, 1, 7, 2, 7, 4, 2, 8, 6, 3, 3, 1, 8, 6, 2, 1, 6, 5, 6, 2, 2, 5, 5, 7, 7, 5, 6, 5, 10, 4, 7, 5

Offset

1,7

Comments

Conjecture: a(n) > 0 for all n > 22. In other words, for any prime p > 80 there is a positive integer k < p/2 such that k + k' is a square, where k' is the unique integer among 1,...,p-1 with k*k' == 1 (mod p).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(54) = 1 since 38*218 is congruent to 1 modulo prime(54)=251 with 38 < 251/2, and 38 + 218 = 16^2 is a square.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[m=0; Do[If[SQ[k+PowerMod[k, -1, Prime[n]]], m=m+1]; Continue, {k, 1, (Prime[n]-1)/2}];
Print[n, " ", m]; Continue, {n, 1, 70}]

Crossrefs

Cf. A000040, A000290.

Sequence in context: A284610 A234017 A182057 * A097027 A072741 A131360

Adjacent sequences: A260925 A260926 A260927 * A260929 A260930 A260931

Keyword

nonn

Author

Zhi-Wei Sun, Aug 04 2015

Status

approved