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

1,1

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)

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.

Formula

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

Example

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.

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 *)

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.

Sequence in context: A332990 A257815 A141655 * A099004 A308007 A360413

Adjacent sequences: A174346 A174347 A174348 * A174350 A174351 A174352

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 16 2010

Extensions

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

Status

approved