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A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals. 2
3, 5, 7, 11, 13, 23, 17, 19, 31, 89, 29, 37, 47, 359, 139, 41, 43, 53, 389, 181, 199, 59, 67, 61, 401, 241, 211, 113, 71, 79, 73, 449, 283, 467, 293, 1831, 101, 97, 83, 479, 337, 509, 317, 1933, 523, 107, 103, 131, 491, 409, 619, 773, 2113, 1069, 887 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Every odd prime p = prime(i), i > 1, occurs in this array, in row (prime(i+1) - prime(i))/2. Polignac's conjecture states that each row contains an infinite number of indices. In case this does not hold, we can use the convention to continue finite rows with 0's, to ensure the sequence is well defined. - M. F. Hasler, Oct 19 2018

LINKS

T. D. Noe, Falling antidiagonals 1..50, flattened

Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.

FORMULA

a(n) = A000040(A174349(n)). - Michel Marcus, Mar 30 2016

EXAMPLE

Corner of the array:

    3     5    11    17    29    41 ...

    7    13    19    37    43    67 ...

   23    31    47    53    61    73 ...

   89   359   389   401   449   479 ...

  139   181   241   283   337   409 ...

  199   211   467   509   619   661 ...

  (...)

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: (lesser of) twin primes A001359.

Row 2: p(4) = 7, p(6) = 13, p(8) = 19,... these being the primes for which the next prime is 4 greater: (lesser of) cousin primes A029710.

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}]; Table[t2[[b, a - b + 1]], {a, rows}, {b, a}] (* T. D. Noe, Feb 11 2014 *)

PROG

(PARI) A174350_row(g, N=50, i=0, p=prime(i+1), L=[])={g*=2; forprime(q=1+p, , i++; if(p+g==p=q, L=concat(L, q-g); N--||return(L)))} \\ Returns the first N terms of row g. - M. F. Hasler, Oct 19 2018

CROSSREFS

Cf. A000040, A001223, A174349.

Rows 1, 2, 3, ...: A001359, A029710, A031924, A031926, A031928 (row 5), A031930, A031932, A031934, A031936, A031938 (row 10), A061779, A098974, A124594, A124595, A124596 (row 15), A126784, A134116, A134117, A134118, A126721 (row 20), A134120, A134121, A134122, A134123, A134124 (row 25), A204665, A204666, A204667, A204668, A126771 (row 30), A204669, A204670.

Rows 35, 40, 45, 50, ...: A204792, A126722, A204764, A050434 (row 50), A204801, A204672, A204802, A204803, A126724 (row 75), A184984, A204805, A204673, A204806, A204807 (row 100); A224472 (row 150).

Column 1: A000230.

Column 2: A046789.

Sequence in context: A119753 A169969 A291175 * A240476 A040140 A066651

Adjacent sequences:  A174347 A174348 A174349 * A174351 A174352 A174353

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 16 2010

EXTENSIONS

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

STATUS

approved

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Last modified September 21 23:55 EDT 2019. Contains 327286 sequences. (Running on oeis4.)