A112052 -id:A112052 - cp's OEIS frontend

A112070 Square array A(x,y) = y-th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, or 0 if no such i exists; read by descending antidiagonals.

3, 5, 7, 11, 9, 23, 13, 15, 25, 49, 19, 17, 47, 71, 121, 21, 31, 73, 119, 311, 169, 27, 33, 95, 191, 551, 479, 289, 29, 39, 97, 239, 671, 1151, 1559, 361, 35, 41, 143, 241, 719, 1319, 2999, 5711, 529, 37, 55, 145, 359, 839, 1679, 3071, 8399, 10559, 841, 43, 57

Offset: 1

Antti Karttunen, Aug 27 2005

This is a permutation of odd numbers greater than unity provided that the sequence A112046 contains only prime values and every prime occurs infinitely many times there. Because the Jacobi symbol is multiplicative with respect to its modulus, it follows that if n occurs on row i and m occurs on row j, then n*m cannot occur before row min(i,j).

			The top left corner of the array:
3,5,11,13,19,21,...
7,9,15,17,31,33,...
23,25,47,73,95,...

A(x, y) = 2*A112060(x, y)+1. Transpose: A112071. Column 1: A112052. Row 1: A047621, Row 2: A112072 Row 3: A112073, Row 4: A112074, Row 5: A112075, Row 6: A112076, Row 7: A112077, Row 8: A112078, Row 9: A112079.

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.

Original entry on oeis.org

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

A112051 a(1)=1, a(n) = first index i (> a(n-1)), where A112046(i) gets a value distinct from any values A112046(1)..A112046(a(n-1)).

Original entry on oeis.org

1, 3, 11, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380, 18240

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Column 1 of A112060 (row 1 of A112061). Cf. A112052.

cp's OEIS Frontend

A112070 Square array A(x,y) = y-th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, or 0 if no such i exists; read by descending antidiagonals.

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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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A112051 a(1)=1, a(n) = first index i (> a(n-1)), where A112046(i) gets a value distinct from any values A112046(1)..A112046(a(n-1)).

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