

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
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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,...,p1 with k*k' == 1 (mod p).


LINKS

ZhiWei 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

ZhiWei Sun, Aug 04 2015


STATUS

approved



