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A192175 Array of primes determined by distance to next prime, by antidiagonals.

%I A192175 #14 Feb 10 2025 03:12:44
%S A192175 2,3,7,5,13,23,11,19,31,89,17,37,47,359,139,29,43,53,389,181,199,41,
%T A192175 67,61,401,241,211,113,59,79,73,449,283,467,293,1831,71,97,83,479,337,
%U A192175 509,317,1933,523,101,103,131,491,409,619,773,2113,1069,887,107
%N A192175 Array of primes determined by distance to next prime, by antidiagonals.
%C A192175 Row 1:  primes p such that p+1 or p+2 is a prime.
%C A192175 Row r>1:  primes p such that the least h for which p+2h is prime is r.
%C A192175 Rows 1-7:  A124588, A023200, A031924, A031926, A031928, A031932, A031924.
%e A192175 Northwest corner:
%e A192175   2.....3.....5.....11....17....29....41
%e A192175   7.....13....19....37....43....67....79
%e A192175   23....31....47....53....61....73....83
%e A192175   89....359...389...401...449...479...491
%e A192175   139...181...241...283...337...409...421
%e A192175 For example, 31 is in row 3 because 31+2*3 is a prime, unlike 31+2*1 and 31+2*2.  Every prime occurs exactly once.  For each row, it is not known whether it is finite.
%t A192175 z = 5000; (* z=number of primes used *)
%t A192175 row[1] = (#1[[1]] &) /@ Cases[Array[{#1,
%t A192175       PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {_, True}];
%t A192175 Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[2 x + Prime[#1]]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 16}]; TableForm[Array[Prime[row[#]] &, {10}]] (* A192175 array *)
%t A192175 Flatten[Table[ Prime[row[k][[n - k + 1]]], {n, 1, 11}, {k, 1, n}]] (* A192175 sequence *)
%t A192175 (* _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192175 Cf. A192176, A192177, A192178, A192179.
%K A192175 nonn,tabl
%O A192175 1,1
%A A192175 _Clark Kimberling_, Jun 24 2011