# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a279967
Showing 1-1 of 1

%I A279967 #24 Jan 23 2017 23:18:36
%S A279967 1,1,2,2,2,7,2,9,10,15,2,10,1,13,17,8,0,13,1,14,9,8,0,13,3,30,13,10,2,
%T A279967 16,1,23,5,7,14,15,2,8,28,32,2,23,2,9,49,12,0,48,2,11,1,20,3,18,13,28,
%U A279967 0,4,1,56,5,8,16,35,46,4,2,6,2,10
%N A279967 Square array read by antidiagonals upwards in which each term is the sum of prior elements in the same row, column, diagonal, or antidiagonal that divide n; the array is seeded with an initial value a(1)=1.
%C A279967 From _Hartmut F. W. Hoft_, Jan 23 2017: (Start)
%C A279967 Shown by induction and direct (modular) computations for
%C A279967 column 1: Every number is even, except for the first two 1's; in addition to row 3, value 2 occurs in rows 4*k and 4*k+1, and every value in rows 4*k+2 and 4*k+3 is divisible by 4, for all k>=1.
%C A279967 column 2: The first four entries, 2, 2, 9 and 10, contain the only odd number; no nonzero entry in row k>3 has 9 as a factor, and value 0 occurs in rows 4*k+1 and 4*k+2, for all k>=1.
%C A279967 Conjecture:
%C A279967 a({1, 6, 8, 9, 10, 15, 26, 45, 48, 84, 96, 112, 115, 252, 336, 343}) =
%C A279967   {1, 7, 9,10, 15, 17, 30, 49, 48,104,117, 115, 122, 257, 343, 395} are the only numbers in the sequence with the property a(n) >= n (verified through n=500500, i.e., the triangle with 1000 antidiagonals).
%C A279967 This conjecture together with Bouniakowsky's conjecture that certain quadratic integer polynomials generate infinitely many primes (e.g. see A002496 for n^2+1 and A188382 for 2*n^2+n+1) implies that in every column in the triangle infinitely many prime sequence indices occur and therefore infinitely many 0's whenever the column contains no 1's. The proof is based on the fact that for a large enough prime sequence index p in whose prior column no 1 occurs then a(p)=0; therefore infinitely many 0's occur in that column. Obviously, once value 1 occurs in a column no 0 value can occur in a subsequent row.
%C A279967 Conjecture:
%C A279967 Every row in the triangle contains exactly two 1's.
%C A279967 (End)
%H A279967 Peter Kagey, <a href="/A279967/b279967.txt">Table of n, a(n) for n = 1..5000</a>
%e A279967 After 6 terms, the array looks like:
%e A279967 .
%e A279967 1   2   7
%e A279967 1   2
%e A279967 2
%e A279967 We have a(6) = 7 because a(1) = 1, a(3) = 2, a(4) = 2, and a(5) = 2 divide 6; 1 + 2 + 2 + 2 = 7.
%e A279967 From _Hartmut F. W. Hoft_, Jan 23 2017: (Start)
%e A279967 1   2   7  15  17   9  10  15  49  13   4  31  22
%e A279967 1   2  10  13  14  13  14   9  18  46  12  66
%e A279967 2   9   1   1  30   7   2   3  35  12   3
%e A279967 2  10  13   3   5  23  20  16  14  17
%e A279967 2   0  13  23   2   1   8  11   2
%e A279967 8   0   1  32  11   5   3   6
%e A279967 8  16  28   2  56  42   8
%e A279967 2   8  48   1   2 104
%e A279967 2   0   4  10   1
%e A279967 12   0   2  10
%e A279967 28   6   2
%e A279967 2  42
%e A279967 2
%e A279967 .
%e A279967 Expanded the triangle to the first 13 antidiagonals of the array, i.e. a(1) ... a(91), to show the start of the 2- and 0-value patterns in columns 1 and 2. The first 0 beyond column 2 is a(677) in row 27, column 11 of the triangle.
%e A279967 A188382(n)=2*n^2+n+1 for n>=0 are the alternate sequence indices for column 1 starting in row 1, 2*n^2+n+2 for n>=1 are the alternate sequence indices for column 2 starting in row 2, and 2*n^2+n+11 for n>=5 are the alternate sequence indices for column 11 starting in row 1.
%e A279967 The sequence indices in the triangle for row positions k>=1 in columns 1,..., 5 are given in sequences A000124(k), A152948(k+3), A152950(k+3), A145018(k+4) and A167499(k+4).
%e A279967 (End)
%t A279967 (*  printing of the triangle is commented out of function a279967[]  *)
%t A279967 pCol[{i_, j_}] := Map[{#, j}&, Range[1, i-1]]
%t A279967 pDiag[{i_, j_}] := If[j>=i, Map[{#, j-i+#}&, Range[1, i-1]], Map[{i-j+#, #}&, Range[1, j-1]]]
%t A279967 pRow[{i_, j_}] := Map[{i, #}&, Range[1, j-1]]
%t A279967 pAdiag[{i_, j_}] := Map[{i+j-#, #}&, Range[1, j-1]]
%t A279967 priorPos[{i_, j_}] := Join[pCol[{i, j}], pDiag[{i, j}], pRow[{i, j}], pAdiag[{i, j}]]
%t A279967 seqPos[{i_, j_}] := (i+j-2)(i+j-1)/2+j
%t A279967 antiDiag[k_] := Map[{k+1-#, #}&, Range[1, k]]
%t A279967 upperTriangle[k_] := Flatten[Map[antiDiag, Range[1, k]], 1]
%t A279967 a279967[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={1}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Apply[Plus, Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; AppendTo[seqL, val]; ms[[Apply[Sequence, pos]]]=val]; (* Print[TableForm[ms]]; *) seqL]
%t A279967 a279967[13] (* values in first 13 antidiagonals *)
%t A279967 (* _Hartmut F. W. Hoft_, Jan 23 2017 *)
%Y A279967 Cf. A279966 for the related sequence which counts prior terms.
%Y A279967 Cf. A269347 for a one-dimensional version of this sequence.
%Y A279967 Cf. also A279211, A279212.
%Y A279967 Cf. A000124, A002496, A145018, A152948, A152950, A167499, A188382.
%K A279967 nonn,tabl
%O A279967 1,3
%A A279967 _Alec Jones_, Dec 24 2016

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE