This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174350 #52 Sep 22 2022 08:13:27
%S A174350 3,5,7,11,13,23,17,19,31,89,29,37,47,359,139,41,43,53,389,181,199,59,
%T A174350 67,61,401,241,211,113,71,79,73,449,283,467,293,1831,101,97,83,479,
%U A174350 337,509,317,1933,523,107,103,131,491,409,619,773,2113,1069,887
%N A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals.
%C A174350 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
%C A174350 A permutation of the odd primes (A065091). - _Robert G. Wilson v_, Sep 13 2022
%H A174350 Robert G. Wilson v, <a href="/A174350/b174350.txt">Falling antidiagonals 1..115, flattened</a> (first 50 from T. D. Noe).
%H A174350 Fred B. Holt and Helgi Rudd, <a href="http://arxiv.org/abs/1402.1970">On Polignac's Conjecture</a>, arxiv:1402.1970 [math.NT], 2014.
%F A174350 a(n) = A000040(A174349(n)). - _Michel Marcus_, Mar 30 2016
%e A174350 Upper left hand corner of the array:
%e A174350 3 5 11 17 29 41 59 71 101 ...
%e A174350 7 13 19 37 43 67 79 97 103 ...
%e A174350 23 31 47 53 61 73 83 131 151 ...
%e A174350 89 359 389 401 449 479 491 683 701 ...
%e A174350 139 181 241 283 337 409 421 547 577 ...
%e A174350 199 211 467 509 619 661 797 997 1201 ...
%e A174350 113 293 317 773 839 863 953 1409 1583 ...
%e A174350 1831 1933 2113 2221 2251 2593 2803 3121 3373 ...
%e A174350 523 1069 1259 1381 1759 1913 2161 2503 2861 ...
%e A174350 (...)
%e A174350 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.
%e A174350 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.
%t A174350 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 A174350 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 *)
%o A174350 (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
%Y A174350 Cf. A000040, A001223, A065091, A174349.
%Y A174350 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.
%Y A174350 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).
%Y A174350 Column 1: A000230.
%Y A174350 Column 2: A046789.
%K A174350 nonn,tabl
%O A174350 1,1
%A A174350 _Clark Kimberling_, Mar 16 2010
%E A174350 Definition corrected and other edits by _M. F. Hasler_, Oct 19 2018