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 A309129 Numbers n such that -n is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n. 0

%I #25 Jul 26 2019 18:29:24

%S 1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,19,21,22,24,25,27,28,30,33,37,40,

%T 42,43,45,48,57,58,60,63,67,70,72,78,85,88,93,100,102,105,112,120,130,

%U 133,135,147,148,163,165,168,177,190,210,232,240,247,253,267,268,273,280,312,330,333,345,357,385,408,462,520,522,652,708,760,840,928,1320,1365,1467,1848

%N Numbers n such that -n is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n.

%C Contains A000926 and A003173 (except the term 11) as subsequences.

%C Conjecture: 1848 is the last term of this sequence.

%e 42 is in this sequence because sqrt(42) = 6.480740..., and -42 is quadratic nonresidue mod all odd primes < 6.480740... not dividing 42 (only mod 5).

%e 67 is in this sequence because sqrt(67) = 8.185352..., and -67 is quadratic nonresidue mod all odd primes < 8.185352... not dividing 67 (mod 3, mod 5 and mod 7).

%e 17 is not in this sequence because -17 is quadratic residue mod 3 and 3 < sqrt(17) and 3 does not divide 17.

%e 90 is not in this sequence because -90 is quadratic residue mod 7 and 7 < sqrt(90) and 7 does not divide 90.

%e For n < 9, the range of p is empty, thus the numbers n < 9 are trivially in this sequence.

%o (PARI) a(n)=forprime(p=3, ,if(kronecker(-n,p)==1,return(p)))

%o for(k=1, 10^6,if(a(k)>sqrt(k),print1(k, ", ")))

%Y Cf. A000926, A003173.

%K nonn

%O 1,2

%A _Richard N. Smith_, Jul 13 2019

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Last modified September 7 18:28 EDT 2024. Contains 375749 sequences. (Running on oeis4.)