A174349 Square array: row n gives the indices i for which prime(i+1) = prime(i) + 2n; read by falling antidiagonals.

2, 3, 4, 5, 6, 9, 7, 8, 11, 24, 10, 12, 15, 72, 34, 13, 14, 16, 77, 42, 46, 17, 19, 18, 79, 53, 47, 30, 20, 22, 21, 87, 61, 91, 62, 282, 26, 25, 23, 92, 68, 97, 66, 295, 99, 28, 27, 32, 94, 80, 114, 137, 319, 180, 154, 33, 29, 36, 124, 82, 121, 146, 331, 205, 259, 189

Offset: 1

Author

Clark Kimberling, Mar 16 2010

Keywords

• nonn
• tabl

Comments

It is conjectured that every positive integer except 1 occurs in the array.

From M. F. Hasler, Oct 19 2018: (Start)

The above conjecture is obviously true: the integer i appears in row (prime(i+1) - prime(i))/2.

Polignac's Conjecture states that all rows are of infinite length.

To ensure the sequence is well-defined in case the conjecture would not hold, we can use the convention that finite rows are continued by 0's. (End)

Examples

			Corner of the array:
   2    3    5    7    10    13 ...
   4    6    8   12    14    17 ...
   9   11   15   16    18    21 ...
  24   72   77   79    87    92 ...
  34   42   53   61    68    80 ...
  46   47   91   97   114   121 ...
  (...)
Row 1: p(2) = 3, p(3) = 5, p(5) = 11, p(7) = 17, ..., these being the primes for which the next prime is 2 greater, cf. A029707.
Row 2: p(4) = 7, p(6) = 13, p(8) = 19, ..., these being the primes for which the next prime is 4 greater, cf. A029709.
		

Links

• T. D. Noe, Falling antidiagonals 1..50 of the array, flattened
• Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.
• Index entries for primes, gaps between.

Crossrefs

Cf. A000040, A000720, A001223, A174350.

Rows 1, 2, 3, ... are A029707, A029709, A320701, ..., A320720; A116493 (row 35), A116496 (row 50), A116497 (row 100), A116495 (row 105).

Column 1 is A038664.

Programs

Mathematica

rows = 10; t2 = {}; Do[t = {}; p = Prime[2]; While[Length[t] < rows - off + 1, nextP = NextPrime[p]; If[nextP - p == 2*off, AppendTo[t, p]]; p = nextP]; AppendTo[t2, t], {off, rows}]; t3 = Table[t2[[b, a - b + 1]], {a, rows}, {b, a}]; PrimePi /@ t3 (* T. D. Noe, Feb 11 2014 *)

Formula

a(n) = A000720(A174350(n)). - Michel Marcus, Mar 30 2016

Extensions

Name corrected and other edits by M. F. Hasler, Oct 19 2018