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A305579 Square array read by antidiagonals upwards in which row k has k as its first term and each subsequent term is the least possible value such that the sum of any 2 or more terms does not equal a prime.

%I A305579 #26 Jul 27 2018 14:40:46
%S A305579 1,2,3,3,4,5,4,5,6,87,5,5,7,8,87,6,7,11,83,10,1151,7,8,9,29,235,12,
%T A305579 5371,8,8,10,79,215,395,14,199276,9,10,13,12,131,511,5275,16,32281747,
%U A305579 10,11,12,37,14,196,8729,76128,18,16946784207,11,11,13,14,67,16,3983,20526,9782734,20
%N A305579 Square array read by antidiagonals upwards in which row k has k as its first term and each subsequent term is the least possible value such that the sum of any 2 or more terms does not equal a prime.
%C A305579 Rows which appear to have consecutive even numbers are for k = 2, 6, 8, 14, 18, 20, 26, 36, 44, 48, 50, 54,56, 68, 74, 78, 86, 96, 114, ..., .
%C A305579 Conjecture: these row terms are a proper subset of A005843.
%e A305579 Row 1 is A133660 and is a good illustration of the definition.
%e A305579 Array begins:
%e A305579 ============================================================================
%e A305579 k\n|  1   2   3   4    5     6      7       8         9           10
%e A305579 ---|------------------------------------------------------------------------
%e A305579 1  |  1,  3,  5, 87, 113, 1151,  5371, 199276, 32281747, 16946784207, ..., ;
%e A305579 2  |  2,  4,  6,  8,  10,   12,    14,     16,       18,          20, ..., ;
%e A305579 3  |  3,  5,  7, 83, 235,  395,  5275,  76128,  9782734, ..., ;
%e A305579 4  |  4,  5, 11, 29, 215,  511,  8729,  20526,  9745499, ..., ;
%e A305579 5  |  5,  7,  9, 79, 131,  196,  3983,  16380,   270270, ..., ;
%e A305579 6  |  6,  8, 10, 12,  14,   16,    18,     20,       22,          24, ..., ;
%e A305579 7  |  7,  8, 13, 37,  67, 1087,  5128, 137886,  6353767, ..., ;
%e A305579 8  |  8, 10, 12, 14,  16,   18,    20,     22,       24,          26, ..., ;
%e A305579 9  |  9, 11, 13, 71, 112,  281,  1952, 147630,  1729159, ..., ;
%e A305579 10 | 10, 11, 14, 25,  94,  756,  2394,  28480,  1466566, ..., ;
%e A305579 11 | 11, 13, 14, 25, 109,  559,  2719,  57985,  2589731, ..., ;
%e A305579 12 | 12, 13, 14, 37,  79,  673,  2929, 113256,  9708060, ..., ;
%e A305579 13 | 13, 14, 19, 31,  97,  882,  2028, 161340,  3635970, ..., ;
%e A305579 14 | 14, 16, 18, 20,  22,   24,    26,     28,       30,          32, ..., ;
%e A305579 15 | 15, 17, 18, 31, 137,  502,  7983, 599346, 27105801, ..., ;
%e A305579 16 | 16, 17, 18, 47, 107,  395,  6480,  91140,   467730, ..., ;
%e A305579 17 | 17, 18, 21, 31,  77,  637,  3609,  77910,   652680, ..., ;
%e A305579 18 | 18, 20, 22, 24,  26,   28,    30,     32,       34,          36, ..., ;
%e A305579 19 | 19, 20, 25, 30,  61,  235,  2965,   4415,   394170,     5769540, ..., ;
%e A305579 20 | 20, 22, 24, 26,  28,   30,    32,     34,       36,          38, ..., ;
%e A305579 21 | 21, 23, 25, 47,  73,  797, 20419, 235665,      ..., ;
%e A305579 22 | 22, 23, 27, 42,  69,  462,   672,    783,    71652,      935298, ..., ;
%e A305579 23 | 23, 25, 26, 37,  73, 1555,  4219, 196260,  3698520, ..., ;
%e A305579 24 | 24, 25, 26, 31, 193,  504,  3756,  91831,  7703843, ..., ;
%e A305579 25 | 25, 26, 29, 31,  39,  750,  4350,  85830,   661350, ..., ;
%e A305579 26 | 26, 28, 30, 32,  34,   36,    38,     40,       42,          44, ..., ;
%e A305579 27 | 27, 28, 29, 35, 232,  888,  5670, 134400,  4058376, ..., ;
%e A305579 28 | 28, 29, 34, 53,  59, 1045,  3696, 249240,  9475589, ..., ;
%e A305579 29 | 29, 31, 33, 55,  57,  674,  6581, 126272,  2549747, ..., ;
%e A305579 30 | 30, 32, 33, 52,  60,   63,    90,    120,      150,         180, ..., ;
%e A305579 31 | 31, 32, 33, 54,  90,  714,  9450, 188850,  2598573, ..., ;
%e A305579 32 | 32, 33, 37, 45, 138,  597,  2703, 101055,  2754885, ..., ;
%e A305579 33 | 33, 35, 37, 47, 133,  555,  4155, 332885,  3090195, ..., ;
%e A305579 34 | 34, 35, 41, 43,  77,  594,  2940,  35700,  2323246, ..., ;
%e A305579 35 | 35, 37, 39, 43, 210, 1061, 10125, 372955, 30373014, ..., ;
%e A305579 36 | 36, 38, 40, 42,  44,   46,    48,     50,       52,          54, ..., ;
%e A305579 37 | 37, 38, 39, 47,  48,  631,  8862, 124851,  4972506, ..., ;
%e A305579 ..., etc.
%t A305579 (* first do *) Needs["Combinatorica`"] (* then *) lst = {k}; g[k_] := Block[{j = 1, l = 2^Length@lst}, While[j < l && !PrimeQ[Plus @@ NthSubset[j, lst] + k], j++ ]; If[j == l, False, True]]; f[n_] := Block[{k = lst[[-1]] + 1}, While[PrimeQ@k || g[k] == True, k++; k++ ]; AppendTo[lst, k]; k]; Do[ Print@ f@ n, {n, 10}] (* _Robert G. Wilson v_, Jun 05 2018 *)
%Y A305579 Cf. A005843, A052349, A133660, A133661, first column: A000027.
%K A305579 nonn,tabl
%O A305579 1,2
%A A305579 _Randy L. Ekl_ and _Robert G. Wilson v_, Jun 05 2018