

A174349


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


23



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.
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 welldefined 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: A295088 A257815 A141655 * A099004 A270194 A055170
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



