A209260 - OEIS

%I #40 Apr 16 2016 23:49:49

%S 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,

%T 36,17,24,39,42,44,45,19,34,40,49,52,54,55,38,51,56,60,63,65,66,23,50,

%U 57,68,72,75,77,78

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

%C 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).

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

%C 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]).

%C 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

%C From _Hartmut F. W. Hoft_, Apr 14 2016: (Start)

%C For a proof of Conjecture 2 see the link.

%C 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)

%H Hartmut F. W. Hoft, <a href="/A209260/b209260.txt">Table of n, a(n) for n = 1..2597</a>, the first 100 rows of the triangle by rows

%H Hartmut F. W. Hoft, <a href="/A209260/a209260_2.pdf">Proof of Conjecture 2</a>

%H L. E. Jeffery, <a href="/A209260/a209260_1.pdf">Proof of Conjecture 1 in A209260</a>.

%e As an irregular triangle:

%e {1};

%e {2, 3};

%e {5, 6};

%e {4, 7, 9, 10};

%e {12, 14, 15};

%e {11, 18, 20, 21};

%e {13, 22, 25, 27, 28};

%e {8, 26, 30, 33, 35, 36};

%e {17, 24, 39, 42, 44, 45};

%e {19, 34, 40, 49, 52, 54, 55};

%e {38, 51, 56, 60, 63, 65, 66};

%e {23, 50, 57, 68, 72, 75, 77, 78};

%e and as the terms appear in their correct positions in A141419:

%e {1                                            };

%e {2,  3                                        };

%e {    5,  6                                    };

%e {4,  7,  9, 10                                };

%e {       12, 14, 15                            };

%e {   11,     18, 20, 21                        };

%e {   13,     22, 25, 27, 28                    };

%e {8,         26, 30, 33, 35, 36                };

%e {   17, 24,         39, 42, 44, 45            };

%e {   19,     34, 40,     49, 52, 54, 55        };

%e {           38,     51, 56, 60, 63, 65, 66    };

%e {   23,         50, 57,     68, 72, 75, 77, 78};

%e 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

%t oddDivs[v_] := Module[{d=Divisors[v]}, Select[Transpose[{d, Reverse[d]}], OddQ[#[[2]]]&]]

%t holes[v_] := Drop[Sort[Map[{#[[1]]+(#[[2]]-1)/2, If[#[[2]]+1<=2*#[[1]], #[[2]], 2*#[[1]]]}&, oddDivs[v]], #1[[1]]<#2[[1]]&], 1]

%t a141419[i_, j_] := j*(2*i-j+1)/2

%t triangle141419[r_] := Table[a141419[i, j], {i, 1, r}, {j, 1, i}]

%t holes209260[r_] := Select[Flatten[Map[holes, Union[Flatten[triangle141419[r]]]], 1], #[[1]]<=r&]

%t 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]

%t a209260[r_] := Select[Flatten[triangle209260[r]], #!=" "&]

%t a209260[12] (* data *)

%t TableForm[triangle209260[12], TableDepth->2] (* triangle with holes *)

%t (* _Hartmut F. W. Hoft_, Apr 14 2016 *)

%o (PARI) try(p)={ \\ Test conjecture at prime p

%o     my(n=p\2);

%o     my(v=vectorsmall(n,i,i),v1=vectorsmall(n-1,i,2*i+1));

%o     while(1,

%o         my(t=#v1-1);

%o         while(t && v1[t]>p, t--);

%o         if(t<1,return(0));

%o         [v1,v]=[vectorsmall(t,i,v1[i]+v1[i+1]-v[i+1]),v1];

%o         for(i=1,#v1,

%o             if(v1[i]==p && isprime(p),return("Conjecture fails at "p))

%o         )

%o     )

%o }; \\ _Charles R Greathouse IV_, Jan 23 2013

%Y Cf. A141419.

%K nonn,tabf

%O 1,2

%A _L. Edson Jeffery_ and _Charles R Greathouse IV_, Jan 23 2013