A112062 -id:A112062 - 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.

A112132 Period 4: repeat [1, 3, 1, 7].

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1

Offset: 1

Antti Karttunen, Aug 28 2005

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

First differences of A112062.

Also half of the first differences of A112072. Cf. A112086.

&cat [[1, 3, 1, 7]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([1, 3, 1, 7]), n=1..50); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 100, {1, 3, 1, 7}] (* Wesley Ivan Hurt, Jul 09 2016 *)

a(n)=1+2*((n-1)%2)*((n-1)%4); \\ Jaume Oliver Lafont, Aug 28 2009; corrected by Antti Karttunen, Mar 31 2013

a(n)=[1,3,1,7][1+(n-1)%4]; \\ Joerg Arndt, Apr 02 2013

A112132(n) = { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],if(1==f[i,2],3,7),1)); }; \\ (implements the multiplicative formula) - Antti Karttunen, May 10 2020

a(n+1) = 3-2*sin(Pi*n/2)-2*(-1)^n. - R. J. Mathar, Oct 08 2011
Multiplicative with a(2) = 3, a(2^e) = 7 if e >= 2, a(p^e) = 1 otherwise. - Antti Karttunen, Mar 31 2013, typo corrected May 02 2020
From Wesley Ivan Hurt, Jul 09 2016: (Start)
G.f.: x*(1+3*x+x^2+7*x^3)/(1-x^4).
a(n) = a(n-4) for n>4.
a(2n) = 5+2*(-1)^n, a(2n-1) = 1. (End)
Dirichlet g.f.: zeta(s)*(1+2^(1-s)+4^(1-s)). - Amiram Eldar, Jan 03 2023

A112072 Odd numbers n for which 3 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.

Original entry on oeis.org

7, 9, 15, 17, 31, 33, 39, 41, 55, 57, 63, 65, 79, 81, 87, 89, 103, 105, 111, 113, 127, 129, 135, 137, 151, 153, 159, 161, 175, 177, 183, 185, 199, 201, 207, 209, 223, 225, 231, 233, 247, 249, 255, 257, 271, 273, 279, 281, 295, 297, 303, 305, 319, 321, 327

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 2 of A112070. a(n) = 2*A112062(n)+1.

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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A112132 Period 4: repeat [1, 3, 1, 7].

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A112072 Odd numbers n for which 3 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.

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