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
Keywords
Examples
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.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Crossrefs
Cf. A273543.
Programs
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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 */
Comments