login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Robert G. Wilson v, Falling antidiagonals 1..115, flattened First 50 from T. D. Noe

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

Upper left hand corner of the array:

     3     5    11    17    29    41    59    71   101 ...

     7    13    19    37    43    67    79    97   103 ...

    23    31    47    53    61    73    83   131   151 ...

    89   359   389   401   449   479   491   683   701 ...

   139   181   241   283   337   409   421   547   577 ...

   199   211   467   509   619   661   797   997  1201 ...

   113   293   317   773   839   863   953  1409  1583 ...

  1831  1933  2113  2221  2251  2593  2803  3121  3373 ...

   523  1069  1259  1381  1759  1913  2161  2503  2861 ...

  (...)

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 *)

t[r_, 0] = 2; t[r_, c_] := Block[{p = NextPrime@ t[r, c - 1], q}, q = NextPrime@ p; While[ p + 2r != q, p = q; q = NextPrime@ q]; p]; Table[ t[r - c + 1, c], {r, 10}, {c, r, 1, -1}] (* Robert G. Wilson v, Nov 06 2020 *)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 26 10:35 EDT 2021. Contains 346294 sequences. (Running on oeis4.)