

A297620


Positive numbers n such that n^2 == p (mod q) and n^2 == q (mod p) for some consecutive primes p,q.


1



6, 10, 12, 24, 42, 48, 62, 72, 76, 84, 90, 93, 108, 110, 120, 122, 128, 145, 146, 174, 187, 188, 194, 204, 208, 215, 220, 228, 232, 240, 241, 264, 297, 306, 310, 314, 317, 326, 329, 336, 349, 357, 366, 372, 386, 408, 410, 423, 426, 431, 444, 454, 456, 468, 470, 474, 518, 522, 535, 538, 546, 548
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OFFSET

1,1


COMMENTS

Positive numbers n such that n^2 == p+q mod (p*q) for some consecutive primes p, q.
Each pair of consecutive primes p,q such that p is a quadratic residue mod q and p and q are not both == 3 (mod 4) contributes infinitely many members to the sequence.
Odd terms of this sequence are 93, 145, 187, 215, 241, 297, 317, 329, 349, 357, 423, 431, 535, ...  Altug Alkan, Jan 01 2018


LINKS

Robert Israel, Table of n, a(n) for n = 1..2305


EXAMPLE

a(3) = 12 is in the sequence because 71 and 73 are consecutive primes with 12^2 == 73 (mod 71) and 12^2 == 71 (mod 73).


MAPLE

N:= 1000: # to get all terms <= N
R:= {}:
q:= 3:
while q < N^2 do
p:= q;
q:= nextprime(q);
if ((p mod 4 <> 3) or (q mod 4 <> 3)) and numtheory:quadres(q, p) = 1 then
xp:= numtheory:msqrt(q, p); xq:= numtheory:msqrt(p, q);
for sp in [1, 1] do for sq in [1, 1] do
v:= chrem([sp*xp, sq*xq], [p, q]);
R:= R union {seq(v+k*p*q, k = 0..(Nv)/(p*q))}
od od;
fi;
od:
sort(convert(R, list));


CROSSREFS

Contains A074924.
Sequence in context: A315139 A046363 A101086 * A074924 A064166 A107371
Adjacent sequences: A297617 A297618 A297619 * A297621 A297622 A297623


KEYWORD

nonn


AUTHOR

Robert Israel and Thomas Ordowski, Jan 01 2018


STATUS

approved



