A209260 Irregular triangular array read by rows, resulting in a permutation of the natural numbers.

1, 2, 3, 5, 6, 4, 7, 9, 10, 12, 14, 15, 11, 18, 20, 21, 13, 22, 25, 27, 28, 8, 26, 30, 33, 35, 36, 17, 24, 39, 42, 44, 45, 19, 34, 40, 49, 52, 54, 55, 38, 51, 56, 60, 63, 65, 66, 23, 50, 57, 68, 72, 75, 77, 78

Offset: 1

Author

L. Edson Jeffery and Charles R Greathouse IV, Jan 23 2013

Keywords

• nonn
• tabf

Comments

One way to derive this sequence row by row is by eliminating from row n of A141419 any integer already present in row m < n of A141419, with the remaining entries comprising row n of this triangle. The resulting irregular triangle then begins as {1}; {2, 3}; {5, 6}; {4, 7, 9, 10}; ... (see the examples below).

If 2*n+1 is a prime, then 2*n+1 appears in row n+1.

Conjecture 1. 2*n+1 is in the set S_n = {union of first n rows of A209260} if and only if 2*n+1 is composite. (Verified for all n <= 10^6 by Charles R Greathouse IV.) This conjecture has been proved (see [Jeffery]).

Conjecture 2. Let N be an integer, N>0, and let D_N be the set of all positive integral divisors of N. Let B_N be the set of all integral solutions of d + (h-1)/2 for which d in D_N, with h=N/d and N=d*h (some solutions will not be integers and so are not in B_N). Then N appears in row r = min(B_N) of A209260. Also, N appears in row b of A141419, for each b in B_N. - L. Edson Jeffery, Feb 13 2013

From Hartmut F. W. Hoft, Apr 14 2016: (Start)

For a proof of Conjecture 2 see the link.

The row-column index pair in A141419 for number v having factorization v = d*h with h odd is: (d+(h-1)/2, h) if h+1 <= 2*d, and (d+(h-1)/2, 2*d) if h+1 > 2*d. The position of number v in this triangle occurs for the minimum value of d+(h-1)/2 among all divisor pairs d,h with v = d*h and h odd. (End)

Examples

			As an irregular triangle:
  {1};
  {2, 3};
  {5, 6};
  {4, 7, 9, 10};
  {12, 14, 15};
  {11, 18, 20, 21};
  {13, 22, 25, 27, 28};
  {8, 26, 30, 33, 35, 36};
  {17, 24, 39, 42, 44, 45};
  {19, 34, 40, 49, 52, 54, 55};
  {38, 51, 56, 60, 63, 65, 66};
  {23, 50, 57, 68, 72, 75, 77, 78};
and as the terms appear in their correct positions in A141419:
  {1                                            };
  {2,  3                                        };
  {    5,  6                                    };
  {4,  7,  9, 10                                };
  {       12, 14, 15                            };
  {   11,     18, 20, 21                        };
  {   13,     22, 25, 27, 28                    };
  {8,         26, 30, 33, 35, 36                };
  {   17, 24,         39, 42, 44, 45            };
  {   19,     34, 40,     49, 52, 54, 55        };
  {           38,     51, 56, 60, 63, 65, 66    };
  {   23,         50, 57,     68, 72, 75, 77, 78};
The row-column index pair for a hole in the triangle can be computed from the expressions in the Formula section and the triangle produced with the Mathematica code. - _Hartmut F. W. Hoft_, Apr 14 2016
		

Links

• Hartmut F. W. Hoft, Table of n, a(n) for n = 1..2597, the first 100 rows of the triangle by rows
• Hartmut F. W. Hoft, Proof of Conjecture 2
• L. E. Jeffery, Proof of Conjecture 1 in A209260.

Crossrefs

Cf. A141419.

Programs

Mathematica

oddDivs[v_] := Module[{d=Divisors[v]}, Select[Transpose[{d, Reverse[d]}], OddQ[#[[2]]]&]]
holes[v_] := Drop[Sort[Map[{#[[1]]+(#[[2]]-1)/2, If[#[[2]]+1<=2*#[[1]], #[[2]], 2*#[[1]]]}&, oddDivs[v]], #1[[1]]<#2[[1]]&], 1]
a141419[i_, j_] := j*(2*i-j+1)/2
triangle141419[r_] := Table[a141419[i, j], {i, 1, r}, {j, 1, i}]
holes209260[r_] := Select[Flatten[Map[holes, Union[Flatten[triangle141419[r]]]], 1], #[[1]]<=r&]
triangle209260[r_] := Module[{rT=triangle141419[r], rH=holes209260[r], k, i, j}, For[k=1, k<=Length[rH], k++, {i, j}=rH[[k]]; rT[[i, j]]=" "]; rT]
a209260[r_] := Select[Flatten[triangle209260[r]], #!=" "&]
a209260[12] (* data *)
TableForm[triangle209260[12], TableDepth->2] (* triangle with holes *)
(* Hartmut F. W. Hoft, Apr 14 2016 *)

PARI

try(p)={ \\ Test conjecture at prime p
    my(n=p\2);
    my(v=vectorsmall(n,i,i),v1=vectorsmall(n-1,i,2*i+1));
    while(1,
        my(t=#v1-1);
        while(t && v1[t]>p, t--);
        if(t<1,return(0));
        [v1,v]=[vectorsmall(t,i,v1[i]+v1[i+1]-v[i+1]),v1];
        for(i=1,#v1,
            if(v1[i]==p && isprime(p),return("Conjecture fails at "p))
        )
    )
}; \\ Charles R Greathouse IV, Jan 23 2013