

A174349


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


24



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
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OFFSET

1,1


COMMENTS

It is conjectured that every positive integer except 1 occurs in the array.
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 welldefined in case the conjecture would not hold, we can use the convention that finite rows are continued by 0's. (End)


LINKS



FORMULA



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

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


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AUTHOR



EXTENSIONS



STATUS

approved



