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

%I A308764 #16 Feb 07 2020 16:41:40
%S A308764 5,7,11,31,0,29,0,23,37,0,139,401,53,97,149,199,0,89,367,0,521,0,467,
%T A308764 337,0,251,223,0,1933,113,509,409,701,1543,127,1949,523,0,953,1201,0,
%U A308764 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
%N A308764 Table read by antidiagonals: T(n,k) is the smallest prime that differs from its predecessor and successor by 2n and 2k, respectively.
%C A308764 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.
%C A308764 Nonzero terms on the main diagonal are the terms of A054342.
%H A308764 Jon E. Schoenfield, <a href="/A308764/b308764.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%F A308764 T(n,k)=0 if n == k (mod 3) !== 0 (mod 3), with the exception of T(1,1)=5.
%e A308764 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).
%e A308764 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).
%e A308764 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.)
%e A308764 Table begins:
%e A308764      5,     7,    31,     0,   139,   199,     0,  1933, ...
%e A308764     11,     0,    23,   401,     0,   467,   113,     0, ...
%e A308764     29,    37,    53,    89,   337,   509,   953,  3643, ...
%e A308764      0,    97,   367,     0,   409,  1201,     0,  1831, ...
%e A308764    149,     0,   251,   701,     0,   797,   293,     0, ...
%e A308764    521,   223,  1543,   479,   631,   211,  2633,  4111, ...
%e A308764      0,   127,   331,     0,   787,  7057,     0, 13381, ...
%e A308764   1949,     0,  3407,  2609,     0,  3659,  1847,     0, ...
%e A308764    ...    ...    ...    ...    ...    ...    ...    ...  ...
%Y A308764 Cf. A000040 (primes), A001223 (prime gaps), A054342 (first occurrence of distances of equidistant lonely primes).
%K A308764 nonn,tabl
%O A308764 1,1
%A A308764 _Jon E. Schoenfield_, Jun 23 2019