A220516 -id:A220516 - cp's OEIS frontend

A220508 T(n,k) = n^2 + k if k <= n, otherwise T(n,k) = k*(k + 2) - n; square array T(n,k) read by ascending antidiagonals (n >= 0, k >= 0).

0, 1, 3, 4, 2, 8, 9, 5, 7, 15, 16, 10, 6, 14, 24, 25, 17, 11, 13, 23, 35, 36, 26, 18, 12, 22, 34, 48, 49, 37, 27, 19, 21, 33, 47, 63, 64, 50, 38, 28, 20, 32, 46, 62, 80, 81, 65, 51, 39, 29, 31, 45, 61, 79, 99, 100, 82, 66, 52, 40, 30, 44, 60, 78, 98, 120

Offset: 0

Omar E. Pol, Feb 09 2013

This sequence consists of 0 together with a permutation of the natural numbers. The nonnegative integers (A001477) are arranged in the successive layers from T(0,0) = 0. The n-th layer start with T(n,1) = n^2. The n-th layer is formed by the first n+1 elements of row n and the first n elements in increasing order of the column n.
The first antidiagonal is formed by odd numbers: 1, 3. The second antidiagonal is formed by even numbers: 4, 2, 8. The third antidiagonal is formed by odd numbers: 9, 5, 7, 15. And so on.
It appears that in the n-th layer there is at least a prime number <= g and also there is at least a prime number > g, where g is the number on the main diagonal, the n-th oblong number A002378(n), if n >= 1.

			The second layer is [4, 5, 6, 7, 8] which looks like this:
  .  .  8
  .  .  7,
  4, 5, 6,
Square array T(0,0)..T(10,10) begins:
    0,   3,   8,  15,  24,  35,  48,  63,  80,  99, 120,...
    1,   2,   7,  14,  23,  34,  47,  62,  79,  98, 119,...
    4,   5,   6,  13,  22,  33,  46,  61,  78,  97, 118,...
    9,  10,  11,  12,  21,  32,  45,  60,  77,  96, 117,...
   16,  17,  18,  19,  20,  31,  44,  59,  76,  95, 118,...
   25,  26,  27,  28,  29,  30,  43,  58,  75,  94, 117,...
   36,  37,  38,  39,  40,  41,  42,  57,  74,  93, 114,...
   49,  50,  51,  52,  53,  54,  55,  56,  73,  92, 113,...
   64,  65,  66,  67,  68,  69,  70,  71,  72,  91, 112,...
   81,  82,  83,  84,  85,  86,  87,  88,  89,  90, 111,...
  100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,...
  ...

Index entries for sequences that are permutations of the natural numbers

Column 1 is A000290. Main diagonal is A002378. Column 2 is essentially A002522. Row 1 is A005563. Row 2 gives the absolute terms of A008865.

Cf. A000005, A000040, A002061, A002620, A014206, A014209, A027688, A028387, A028552, A060736, A220516.

From Petros Hadjicostas, Mar 10 2021: (Start)
T(n,k) = (A342354(n,k) - 1)/2.
O.g.f.: (x^4*y^3 + 3*x^3*y^4 + x^4*y^2 - 10*x^3*y^3 - x^2*y^4 + 3*x^3*y^2 + x^2*y^3 - 4*x^3*y + 8*x^2*y^2 + 3*x^2*y + x*y^2 + x^2 - 10*x*y - y^2 + x + 3*y)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2). (End)

Name edited by Petros Hadjicostas, Mar 10 2021

A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 5, 3, 9, 16, 6, 8, 10, 25, 15, 7, 11, 17, 26, 24, 14, 12, 18, 36, 49, 27, 23, 13, 19, 35, 37, 64, 48, 28, 22, 20, 34, 38, 50, 65, 63, 47, 29, 21, 33, 39, 51, 81, 100, 66, 62, 46, 30, 32, 40, 52, 80, 82, 121, 99, 67, 61, 45, 31, 41, 53, 79, 83, 101

Offset: 1

Boris Putievskiy, Mar 06 2013

Permutation of the natural numbers. a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.In general, let b(z) be a sequence of integer numbers. Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Natural numbers placed in table T(n,k) layer by layer. The order of placement - layer is counterclockwise, if b(z) is odd; layer is clockwise if b(z) is even. T(n,k) read by antidiagonals.For A219159 - the order of the placement - at the beginning m layers counterclockwise, next m layers clockwise and so on - b(z)=floor((z-1)/m)+1. For this sequence b(z)=z^2 mod 3.

			The start of the sequence as table.
The direction of the placement denotes by ">" and  "v".
  ..........v...........v...........v
  >1....4...5..16..25..26..49..64..65...
  >2....3...6..15..24..27..48..63..66...
  .9....8...7..14..23..28..47..62..67...
  >10..11..12..13..22..29..46..61..68...
  >17..18..19..20..21..30..45..60..69...
  .36..35..34..33..32..31..44..59..70...
  >37..38..39..40..41..42..43..58..71...
  >50..51..52..53..54..55..56..57..72...
  .81..80..79..78..77..76..75..74..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  4,2;
  5,3,9;
  16,6,8,10;
  25,15,7,11,17;
  26,24,14,12,18,36;
  49,27,23,13,19,35,37;
  64,48,28,22,20,34,38,50;
  65,63,47,29,21,33,39,51,81;
  . . .

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A219159, A081344, A194280, A042964, A130196, A011655, A220516.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j>=i:
   result=((1+(-1)**(j**2%3-1))*(j**2-i+1)-(-1+(-1)**(j**2%3-1))*((j-1)**2 +i))/2
else:
   result=((1+(-1)**(i**2%3))*(i**2-j+1)-(-1+(-1)**(i**2%3))*((i-1)**2 +j))/2

For general case.
As table
T(n,k) = ((1+(-1)^(b(k)-1))*(k^2-n+1)-(-1+(-1)^(b(k)-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(b(j)-1))*(j^2-i+1)-(-1+(-1)^(b(j)-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, if  i >j;
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
For this sequence b(z)=z^2 mod 3.
As table
T(n,k) = ((1+(-1)^(k^2 mod 3-1))*(k^2-n+1)-(-1+(-1)^(k^2 mod 3-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^(n^2 mod 3))*(n^2-k+1)-(-1+(-1)^(n^2 mod 3))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(j^2 mod 3-1))*(j^2-i+1)-(-1+(-1)^(j^2 mod 3-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^(i^2 mod 3))*(i^2-j+1)-(-1+(-1)^(i^2 mod 3))*((i-1)^2 +j))/2, if  i >j;
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A222030 Primes and quarter-squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 19, 20, 23, 25, 29, 30, 31, 36, 37, 41, 42, 43, 47, 49, 53, 56, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 90, 97, 100, 101, 103, 107, 109, 110, 113, 121, 127, 131, 132, 137, 139, 144, 149, 151, 156, 157, 163, 167, 169

Offset: 0

Omar E. Pol, Feb 05 2013

Union of A002620 and A000040.
It appears that there is always a prime between two consecutive quarter squares, if n >= 2. Therefore in a square spiral, or zig-zag, whose vertices are the quarter-squares, it appears that there is always a prime between two consecutive vertices, if n >= 2.
Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013
Union of A000040 and A000290 and A002378. - Omar E. Pol, Oct 28 2013

Wikipedia, Oppermann's conjecture

Cf. A000040, A002620, A000290, A014085, A220492, A220506, A220508, A220516.

mx = 13; Union[Prime[Range[PrimePi[mx^2]]], Floor[Range[2*mx]^2/4]] (* Alonso del Arte, Mar 03 2013 *)

a(n) ~ n log n. - Charles R Greathouse IV, Mar 04 2013

cp's OEIS Frontend

A220508 T(n,k) = n^2 + k if k <= n, otherwise T(n,k) = k*(k + 2) - n; square array T(n,k) read by ascending antidiagonals (n >= 0, k >= 0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A222030 Primes and quarter-squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula