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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
A permutation of the odd primes (A065091). - Robert G. Wilson v, Sep 13 2022
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
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
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 February 24 14:59 EST 2024. Contains 370305 sequences. (Running on oeis4.)