A227198 -id:A227198 - cp's OEIS frontend

A227196 a(n) = first i >= 1 for which the Kronecker symbol K(i,n) is not +1 (i.e., is either 0 or -1), 0 if no such i exists.

0, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 5, 2, 5, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 5, 2, 7, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 7, 2, 5, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3

Offset: 1

Antti Karttunen, Jul 06 2013

nonn

a(1) = 0, because K(i,1) is 1 for all i. After that, A112046 interleaved with A007395.

All terms beyond a(1) = 0 are prime numbers. Heuristically a(n) is 2 3/4 of the time, 3 1/6 of the time, 5 1/20 of the time, 7 2/105 of the time, etc. The average value is 2.5738775742512.... - Charles R Greathouse IV, Jan 30 2018

A.H.M. Smeets, Table of n, a(n) for n = 1..20163 (first 87 terms from Antti Karttunen)

Bisections: A112046 (for odd terms from 3 onward), A007395 (all even terms).

Cf. A227195, A227197, A227198, A112052.

a(n) = for(k=1,n,if(kronecker(k,n)<1, return(k)))
for(n=1,120, print1(a(n),", "))

A227195(n) = a(n)-1 for all n>=2.

a(2n+1) = A112046(n) for all n>0. - A.H.M. Smeets Jan 29 2018

A227197 Integers k for which the first non-one value of Kronecker symbols K(1,k), K(2,k), ..., K(u,k) is 0, not -1.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106

Offset: 1

Antti Karttunen, Jul 06 2013

nonn

Contains all even numbers >= 2. A227198 gives the odd terms.

Gives all k such that A020639(k) = A227196(k).

Cf. A227196, A227198.

A227196(n) = for(k=1,n,if(kronecker(k,n)<1,return(k)))
for(n=2,500,if((0==kronecker(A227196(n),n)),print1(n,", ")))

A227194 Natural numbers for which all sums Sum_{i=1..u} K(i/n) (with u ranging from 1 to n) are nonnegative, where K(i/n) is Kronecker symbol of i and n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 23, 24, 25, 26, 27, 30, 31, 34, 35, 36, 38, 39, 42, 44, 47, 48, 49, 54, 55, 56, 58, 59, 62, 63, 64, 66, 71, 74, 75, 76, 79, 81, 82, 83, 86, 87, 94, 95, 96, 100, 103, 106, 108, 110, 111, 114, 119, 120, 121, 130, 131

Offset: 1

Antti Karttunen, Jul 06 2013

nonn

Subsets: A095100, A095102 (gives the prime terms). Cf. A227196-A227198.

firstdiving(n) = {s=0;for(k=1,n,s=s+kronecker(k,n);if(s<0,return(k)));return(0)}
for(n=1,500,if((0==firstdiving(n)),print1(n,", ")))

cp's OEIS Frontend

A227196 a(n) = first i >= 1 for which the Kronecker symbol K(i,n) is not +1 (i.e., is either 0 or -1), 0 if no such i exists.

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A227197 Integers k for which the first non-one value of Kronecker symbols K(1,k), K(2,k), ..., K(u,k) is 0, not -1.

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A227194 Natural numbers for which all sums Sum_{i=1..u} K(i/n) (with u ranging from 1 to n) are nonnegative, where K(i/n) is Kronecker symbol of i and n.

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PARI