A249692 -id:A249692 - cp's OEIS frontend

A249778 a(1)=2; thereafter, a(n) is the smallest number not occurring earlier such that Kronecker(a(k), a(n)) = 1 for the next n indices k = n+1, n+2, ..., 2n.

2, 7, 3, 4, 11, 25, 9, 19, 16, 49, 17, 59, 67, 121, 81, 169, 43, 115, 64, 191, 293, 361, 289, 587, 269, 841, 863, 961, 1031, 1369, 441, 1681, 1867, 2209, 1849, 2809, 65, 529, 256, 643, 3481, 3517, 1639, 1877, 3721, 4489, 5041, 5329, 5591, 6241, 3557, 6889, 7921

Offset: 1

Michel Lagneau, Nov 05 2014

nonn

Kronecker(i,j) is an extension of the Jacobi symbol to all integers. The sequence with the condition Kronecker(a(i), a(n)) = -1/+1 is given by A247665.

			a(1) = 2 because the next term is 7 and k(2,7) = 1;
a(2) = 7 because the next two terms are (3,4) => k(7,3) = 1 and k(7,4) = 1;
a(3) = 3 because the next three terms are (4,11,25) => k(3,4)= 1, k(3,11) = 1 and k(3,25) = 1.

Cf. A247665, A249692.

m=55; v=vector(m); u=vectorsmall(1000*m); for(n=1, m, for(i=2, 10^9, if(!u[i], for(j=(n+1)\2, n-1, if(kronecker(v[j], i)==-1 || kronecker(v[j], i)==0, next(2))); v[n]= i; u[i]=1; break))); v

A249782 a(1)=2; thereafter, a(n) is the smallest prime not yet used which is compatible with the condition that a(n) is a quadratic residue modulo a(k) for the next n indices k = n+1, n+2, ..., 2n.

Original entry on oeis.org

2, 7, 3, 37, 11, 83, 107, 139, 43, 211, 193, 881, 751, 2777, 2633, 5981, 6563, 10531, 3407, 9871, 12421, 23873, 6449, 164789, 3547, 39877, 248909, 370081, 528883, 1144453, 574813, 1201153, 1428929, 2225053, 1397719

Offset: 1

Michel Lagneau, Nov 05 2014

L(a(n)/a(k)) = 1 for the next n indices k = n+1, n+2, ..., 2n where L(a/p) is the Legendre symbol.

			a(1) = 2 because the next term is 7 and L(2/7) = 1;
a(2) = 7 because the next two terms are (3,37) => L(7/3) = 1 and L(7/37) = 1;
a(3) = 3 because the next three terms are (37,11,83) => L(3/37) = 1, L(3/11) = 1 and L(3/83) = 1.

Cf. A249692, A249778.

m=35; v=vector(m); u=vectorsmall(10000*m); for(n=1, m, for(i=1, 10^9, if(!u[i], for(j=(n+1)\2, n-1, if(kronecker(v[j], prime(i))==-1 | | kronecker(v[j],prime(i))==0, next(2))); v[n]=prime(i); u[i]=1; break))); v

cp's OEIS Frontend

A249778 a(1)=2; thereafter, a(n) is the smallest number not occurring earlier such that Kronecker(a(k), a(n)) = 1 for the next n indices k = n+1, n+2, ..., 2n.

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