A274127 Numbers n for which 9 is a nontrivial quadratic residue mod n but 9 has only trivial quadratic residues +/- 3 mod each of the proper divisors of n.

16, 18, 20, 24, 27, 28, 35, 44, 45, 52, 55, 60, 63, 65, 68, 70, 76, 77, 84, 85, 91, 92, 95, 99, 100, 105, 110, 115, 116, 117, 119, 124, 130, 132, 133, 140, 143, 145, 148, 153, 154, 155, 156, 161, 164, 165, 170, 171, 172, 175, 182, 185, 187, 188, 190, 195, 196, 203

Offset

1,1

Comments

This sequence is a subsequence of A273543; for m,k > 0, a(m) * k is in A273543. All terms of A273543 are of that form. In other words, these are the primitive terms of A273543.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

x^2 == 9 (mod 16) has solutions 3, 5, 11 and 13, so x^2 == 9 (mod 16 * k) have these solutions, and possibly more as well. Only solutions to the equation x^2 == 9 (mod n) are shown where no proper divisor of n has nontrivial solutions. So 32 isn't shown as 16 has nontrivial solutions as well and is a proper divisor of 32.

Prog

(PARI) is(n, {t=1})={if(n<16, return(0)); my(v2=valuation(n, 2), v3=valuation(n, 3), k=n/2^v2/3^v3); if(v2<3 && v3<2, if(v2>1, k>1, !isprimepower(k)), if(t==1, d=divisors(n); sum(i=1, #d-1, is(d[i], 0))==0, 1))} /* David A. Corneth, Jun 11 2016, modified PARI-program in A273543 from Charles R Greathouse IV */

Crossrefs

Cf. A273543.

Sequence in context: A031315 A077914 A273543 * A328736 A348143 A161874

Adjacent sequences: A274124 A274125 A274126 * A274128 A274129 A274130

Keyword

nonn

Author

David A. Corneth, Jun 10 2016

Status

approved