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

%I A174349 #19 Oct 24 2018 19:16:03
%S A174349 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,
%T A174349 53,47,30,20,22,21,87,61,91,62,282,26,25,23,92,68,97,66,295,99,28,27,
%U A174349 32,94,80,114,137,319,180,154,33,29,36,124,82,121,146,331,205,259,189
%N A174349 Square array: row n gives the indices i for which prime(i+1) = prime(i) + 2n; read by falling antidiagonals.
%C A174349 It is conjectured that every positive integer except 1 occurs in the array.
%C A174349 From _M. F. Hasler_, Oct 19 2018: (Start)
%C A174349 The above conjecture is obviously true: the integer i appears in row (prime(i+1) - prime(i))/2.
%C A174349 Polignac's Conjecture states that all rows are of infinite length.
%C A174349 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)
%H A174349 T. D. Noe, <a href="/A174349/b174349.txt">Falling antidiagonals 1..50 of the array, flattened</a>
%H A174349 Fred B. Holt and Helgi Rudd, <a href="http://arxiv.org/abs/1402.1970">On Polignac's Conjecture</a>, arxiv:1402.1970 [math.NT], 2014.
%H A174349 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.
%F A174349 a(n) = A000720(A174350(n)). - _Michel Marcus_, Mar 30 2016
%e A174349 Corner of the array:
%e A174349    2    3    5    7    10    13 ...
%e A174349    4    6    8   12    14    17 ...
%e A174349    9   11   15   16    18    21 ...
%e A174349   24   72   77   79    87    92 ...
%e A174349   34   42   53   61    68    80 ...
%e A174349   46   47   91   97   114   121 ...
%e A174349   (...)
%e A174349 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.
%e A174349 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.
%t A174349 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 *)
%Y A174349 Cf. A000040, A000720, A001223, A174350.
%Y A174349 Rows 1, 2, 3, ... are A029707, A029709, A320701, ..., A320720; A116493 (row 35), A116496 (row 50), A116497 (row 100), A116495 (row 105).
%Y A174349 Column 1 is A038664.
%K A174349 nonn,tabl
%O A174349 1,1
%A A174349 _Clark Kimberling_, Mar 16 2010
%E A174349 Name corrected and other edits by _M. F. Hasler_, Oct 19 2018