A308764 Table read by antidiagonals: T(n,k) is the smallest prime that differs from its predecessor and successor by 2n and 2k, respectively.

5, 7, 11, 31, 0, 29, 0, 23, 37, 0, 139, 401, 53, 97, 149, 199, 0, 89, 367, 0, 521, 0, 467, 337, 0, 251, 223, 0, 1933, 113, 509, 409, 701, 1543, 127, 1949, 523, 0, 953, 1201, 0, 479, 331, 0, 1277, 0, 2861, 3643, 0, 797, 631, 0, 3407, 1087, 0, 1951, 887, 1069, 1831, 293, 211, 787, 2609, 541, 907, 1151

Offset: 1

Author

Jon E. Schoenfield, Jun 23 2019

Keywords

• nonn
• tabl

Comments

If two consecutive primes p and q appear in the table, then the column number in which p appears is the row number in which q appears. E.g., 23 is in column 3 and 29 is in row 3, 29 is in column 1 and 31 is in row 1, 113 is in column 7 and 127 is in row 7, 3643 is in column 8 and 3659 is in row 8.

Nonzero terms on the main diagonal are the terms of A054342.

Examples

			T(1,1)=5 because 5 is the only prime p whose predecessor and successor primes are p-2 and p+2, respectively (i.e., 3 and 7).
T(7,2)=127 because 127 is the smallest prime p whose predecessor and successor primes are p-14 and p+4, respectively (i.e., 113 and 131).
T(2,2)=0: the only set of three numbers {p-4, p, p+4} that are all prime is the set {3, 7, 11}, but these are not consecutive primes. (For every set of three integers {m-4, m, m+4}, exactly one of the three is divisible by 3.)
Table begins:
     5,     7,    31,     0,   139,   199,     0,  1933, ...
    11,     0,    23,   401,     0,   467,   113,     0, ...
    29,    37,    53,    89,   337,   509,   953,  3643, ...
     0,    97,   367,     0,   409,  1201,     0,  1831, ...
   149,     0,   251,   701,     0,   797,   293,     0, ...
   521,   223,  1543,   479,   631,   211,  2633,  4111, ...
     0,   127,   331,     0,   787,  7057,     0, 13381, ...
  1949,     0,  3407,  2609,     0,  3659,  1847,     0, ...
   ...    ...    ...    ...    ...    ...    ...    ...  ...
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Crossrefs

Cf. A000040 (primes), A001223 (prime gaps), A054342 (first occurrence of distances of equidistant lonely primes).

Formula

T(n,k)=0 if n == k (mod 3) !== 0 (mod 3), with the exception of T(1,1)=5.