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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)). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 13 20:25 EDT 2021. Contains 345009 sequences. (Running on oeis4.)