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%I #36 Aug 09 2017 22:16:31
%S 2,3,4,5,6,5,7,9,28,30,11,12,32,60,45,13,15,53,68,64,97,17,18,58,85,
%T 130,223,160,19,21,62,116,193,322,558,359,23,26,74,144,208,401,868,
%U 713,536,29,30,96,165,238,540,957,1180,1553,2866,31,34,136,186,265,576,1403
%N T(1, c) = prime(c). T(r + 1, c) = (T(r, c') + T(r, c'+1)) / 2 where c' is the c-th number such that T(r, c') + T(r, c'+1) is even. Table for T read downwards by antidiagonals.
%C This array has the same idea as Gilbreath's conjecture (see A036262) but instead of absolute difference it is the integer average sum.
%H Robert G. Wilson v, <a href="/A287943/b287943.txt">Table of n, a(n) for n = 1..465</a>
%e Row
%e 1: 2 3 5 7 11 13 17
%e 2: 4 6 9 12 15 18 21
%e 3: 5 28 32 53 58 62 74
%e 4: 30 60 68 85 116 144 165
%e 5: 45 64 130 193 208 238 265
%e 6: 97 223 322 401 540 576 765
%e 7: 160 558 868 957 1403 1531 1598
%e 8: 359 713 1180 1467 1639 1808 3131
%e 9: 536 1553 4179 5178 6335 7865 9274
%e 10: 2866 7100 14023 14900 15838 17837 20121
%e 11: 4983 15369 18979 22054 28390 43704 47511
%e 12: 10176 17174 25222 36047 60602 87739 120599
%e 13: 13675 21198 104169 155638 193710 201367 223740
%e 14: 174674 271986 372056 479130 542177 553224 581451
%e 15: 223330 322021 425593 590611 650029 807687 924065
%e 16: 373807 508102 620320 728858 865876 1094922 1133312
%e 17: 564211 674589 797367 980399 1114117 1378160 2055687
%e 18: 619400 735978 888883 1047258 3000375 4135480 5526718
%e 19: 677689 4831099 5819401 7119393 7743933 8367375 9362587
%e 20: 2754394 5325250 6469397 7431663 8055654 8864981 14204980
%e 21: 4039822 6950530 36789607 41026156 43928115 47881364 50592342
%e 22: 5495176 49236853 51408848 61276421 64658379 88092051 96453019
%e 23: 62967400 76375215 92272535 119006122 209296919 261901315 310000824
%e 24: 84323875 235599117 316302735 400483922 497171955 515469235 524697491
%e 25: 159961496 275950926 506320595 520083363 555977282 619254662 638646183
%e 26: 217956211 513201979 587615972 647540001 684757327 812990322 1671545118
%e 27: 365579095 666148664 1242267720 1989912374 2194765721 2371664980 2708581740
%e 28: 954208192 1616090047 2540123360 3262521514 3383785254 3840848685
%e 29: 2901322437 3323153384
%e etc.
%e The 2nd row begins with 4, 6 and 9 since it is the integer average, 4 is the average between 3 and 5, six is the average between 5 and 7, and nine is the average between 7 and 11, etc.
%t t = NestList[Select[(Rest@# + Most@#)/2, IntegerQ] &, Prime@ Range@ 1100, 10]; Table[ t[[n -k +1, k]], {n, 11}, {k, n, 1, -1}] // Flatten
%Y Cf. A000040, A024675, A036262.
%K nonn,tabl
%O 1,1
%A _Zak Seidov_ and _Robert G. Wilson v_, Jun 03 2017