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Array of primes determined by distance to next prime, by antidiagonals.
5

%I #14 Feb 10 2025 03:12:44

%S 2,3,7,5,13,23,11,19,31,89,17,37,47,359,139,29,43,53,389,181,199,41,

%T 67,61,401,241,211,113,59,79,73,449,283,467,293,1831,71,97,83,479,337,

%U 509,317,1933,523,101,103,131,491,409,619,773,2113,1069,887,107

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

%C Row 1: primes p such that p+1 or p+2 is a prime.

%C Row r>1: primes p such that the least h for which p+2h is prime is r.

%C Rows 1-7: A124588, A023200, A031924, A031926, A031928, A031932, A031924.

%e Northwest corner:

%e 2.....3.....5.....11....17....29....41

%e 7.....13....19....37....43....67....79

%e 23....31....47....53....61....73....83

%e 89....359...389...401...449...479...491

%e 139...181...241...283...337...409...421

%e 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 z = 5000; (* z=number of primes used *)

%t row[1] = (#1[[1]] &) /@ Cases[Array[{#1,

%t PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {_, True}];

%t 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 Flatten[Table[ Prime[row[k][[n - k + 1]]], {n, 1, 11}, {k, 1, n}]] (* A192175 sequence *)

%t (* _Peter J. C. Moses_, Jun 20 2011 *)

%Y Cf. A192176, A192177, A192178, A192179.

%K nonn,tabl,changed

%O 1,1

%A _Clark Kimberling_, Jun 24 2011