A227196 -id:A227196 - cp's OEIS frontend

A112060 Square array A(x,y) = y-th natural number k for which A112049(k)=x and 0 if no such k exists; read by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ...

1, 2, 3, 5, 4, 11, 6, 7, 12, 24, 9, 8, 23, 35, 60, 10, 15, 36, 59, 155, 84, 13, 16, 47, 95, 275, 239, 144, 14, 19, 48, 119, 335, 575, 779, 180, 17, 20, 71, 120, 359, 659, 1499, 2855, 264, 18, 27, 72, 179, 419, 839, 1535, 4199, 5279, 420, 21, 28, 83, 204, 504

Offset: 1

Antti Karttunen, Aug 27 2005

This is a permutation of natural numbers provided that the sequence A112046 contains only prime values [which is true] and every prime occurs infinitely many times there.

			The top left corner of the array:
   1,  2,  5,  6,  9, 10, ...
   3,  4,  7,  8, 15, 16, ...
  11, 12, 23, 36, 47, ...

Index entries for sequences that are permutations of the natural numbers

A112070(x, y) = 2*A(X, Y)+1. Transpose: A112061. Column 1: A112051. Row 1: A042963, Row 2: A112062, Row 3: A112063, Row 4: A112064, Row 5: A112065, Row 6: A112066, Row 7: A112067, Row 8: A112068, Row 9: A112069.

Cf. also A227196.

A227198 Odd terms in A227197.

Original entry on oeis.org

9, 15, 25, 33, 39, 49, 57, 63, 81, 87, 95, 105, 111, 119, 121, 129, 135, 145, 153, 159, 169, 177, 183, 201, 207, 215, 225, 231, 249, 255, 265, 273, 279, 289, 297, 303, 321, 327, 335, 345, 351, 361, 369, 375, 385, 393, 399, 417, 423, 441, 447, 455, 465, 471, 489

Offset: 1

Antti Karttunen, Jul 06 2013

nonn

Gives all terms n for which A090368((n+1)/2) = A112046((n-1)/2).

Contains all odd squares. What else?

Cf. A227196, A227197, A090368, A112046.

A227196(n) = for(k=1, n, if(kronecker(k, n)<1, return(k)))
for(n=2,1000,if((0==kronecker(A227196(n),n)&&1==(n%2)),print1(n,", ")))

A227195 a(n) = one less than the first i >= 1 for which the Kronecker symbol K(i,n) is either 0 or -1. Or the maximal value u such that all K(1,n), K(2,n), ... up to K(u,n) have +1 as their values.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2

Offset: 2

Antti Karttunen, Jul 06 2013

nonn

A112050 interleaved with A000012.

Antti Karttunen, Table of n, a(n) for n = 2..10000

Bisections: A112050 (for odd terms from 3 onward), A000012 (for all even terms).

A227195(n) = for(k=2, n, if(kronecker(k, n)<1, return(k-1))); \\ Antti Karttunen, Nov 08 2018

a(n) = A227196(n)-1 for all n >= 2.

Data section extended up to a(105) by Antti Karttunen, Nov 08 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

A112060 Square array A(x,y) = y-th natural number k for which A112049(k)=x and 0 if no such k exists; read by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ...

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A227198 Odd terms in A227197.

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A227195 a(n) = one less than the first i >= 1 for which the Kronecker symbol K(i,n) is either 0 or -1. Or the maximal value u such that all K(1,n), K(2,n), ... up to K(u,n) have +1 as their values.

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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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