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Revision History for A279967 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
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.
(history; published version)
#24 by N. J. A. Sloane at Mon Jan 23 23:18:36 EST 2017
STATUS

proposed

approved

#23 by Omar E. Pol at Mon Jan 23 14:18:38 EST 2017
STATUS

editing

proposed

#22 by Omar E. Pol at Mon Jan 23 14:18:04 EST 2017
COMMENTS

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.

EXAMPLE

.

From Hartmut F. W. Hoft, Jan 23 2017: (Start)

From Hartmut F. W. Hoft, Jan 23 2017: (Start)

Discussion
Mon Jan 23
14:18
Omar E. Pol: Minor edits.
#21 by Omar E. Pol at Mon Jan 23 14:15:25 EST 2017
EXAMPLE

1 2 7

1 2

2

1 2 7 15 17 9 10 15 49 13 4 31 22

1 2 10 13 14 13 14 9 18 46 12 66

2 9 1 1 30 7 2 3 35 12 3

2 10 13 3 5 23 20 16 14 17

2 0 13 23 2 1 8 11 2

8 0 1 32 11 5 3 6

8 16 28 2 56 42 8

2 8 48 1 2 104

2 0 4 10 1

2 42

2

CROSSREFS

Cf. A000124, A002496, A145018, A152948, A152950, A167499, A188382. _Hartmut F. W. Hoft_, Jan 23 2017

STATUS

proposed

editing

#20 by Hartmut F. W. Hoft at Mon Jan 23 12:24:45 EST 2017
STATUS

editing

proposed

#19 by Hartmut F. W. Hoft at Mon Jan 23 12:21:22 EST 2017
COMMENTS

From Hartmut F. W. Hoft, Jan 23 2017: (Start)

Shown by induction and direct (modular) computations for

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.

column 2: The first four entries, 2, 2, 9 & 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.

Conjecture:

a({1, 6, 8, 9, 10, 15, 26, 45, 48, 84, 96, 112, 115, 252, 336, 343}) =

{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).

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.

Conjecture:

Every row in the triangle contains exactly two 1's.

(End)

EXAMPLE

1 2 7

1 2

2

.

1 2 7 15 17 9 10 15 49 13 4 31 22

1 2 10 13 14 13 14 9 18 46 12 66

2 9 1 1 30 7 2 3 35 12 3

2 10 13 3 5 23 20 16 14 17

2 0 13 23 2 1 8 11 2

8 0 1 32 11 5 3 6

8 16 28 2 56 42 8

2 8 48 1 2 104

2 0 4 10 1

12 0 2 10

28 6 2

2 42

2

.

From Hartmut F. W. Hoft, Jan 23 2017: (Start)

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.

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.

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).

(End)

MATHEMATICA

(* printing of the triangle is commented out of function a279967[] *)

pCol[{i_, j_}] := Map[{#, j}&, Range[1, i-1]]

pDiag[{i_, j_}] := If[j>=i, Map[{#, j-i+#}&, Range[1, i-1]], Map[{i-j+#, #}&, Range[1, j-1]]]

pRow[{i_, j_}] := Map[{i, #}&, Range[1, j-1]]

pAdiag[{i_, j_}] := Map[{i+j-#, #}&, Range[1, j-1]]

priorPos[{i_, j_}] := Join[pCol[{i, j}], pDiag[{i, j}], pRow[{i, j}], pAdiag[{i, j}]]

seqPos[{i_, j_}] := (i+j-2)(i+j-1)/2+j

antiDiag[k_] := Map[{k+1-#, #}&, Range[1, k]]

upperTriangle[k_] := Flatten[Map[antiDiag, Range[1, k]], 1]

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]

a279967[13] (* values in first 13 antidiagonals *)

(* Hartmut F. W. Hoft, Jan 23 2017 *)

KEYWORD

nonn,tabl

STATUS

approved

editing

Discussion
Mon Jan 23
12:24
Hartmut F. W. Hoft: I entered the entire triangle of 13 anti diagonals and I added keyword tabl
#18 by N. J. A. Sloane at Tue Dec 27 22:47:09 EST 2016
STATUS

editing

approved

#17 by N. J. A. Sloane at Tue Dec 27 22:47:07 EST 2016
NAME

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.

STATUS

proposed

editing

#16 by Peter Kagey at Mon Dec 26 19:41:26 EST 2016
STATUS

editing

proposed

Discussion
Tue Dec 27
01:15
Hartmut F. W. Hoft: Why is a(1)=1? There are no "prior" elements to the element in the upper left hand corner of the matrix in my interpretation of the term. What am I missing?
01:38
Peter Kagey: I think the sequence should be defined as "a(1) = 1, and ..." otherwise this would just be the zero sequence.
21:04
Hartmut F. W. Hoft: correct, but then a(1)=1 ought to be stated in the Name of the sequence since it is an element (the only one) that does not satisfy the rule for all other values of the sequence. This should then apply to A279966 also.
21:22
Peter Kagey: I agree. I'd support an editor making this change.
#15 by Peter Kagey at Mon Dec 26 19:40:52 EST 2016
LINKS

Peter Kagey, <a href="/A279967/b279967.txt">Table of n, a(n) for n = 1..5000</a>

STATUS

approved

editing