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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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
Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.
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
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 16 2010
EXTENSIONS
Name corrected and other edits by M. F. Hasler, Oct 19 2018
STATUS
approved

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Last modified June 19 05:57 EDT 2024. Contains 373492 sequences. (Running on oeis4.)