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A240062
Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts.
21
1, 2, 3, 4, 5, 9, 6, 7, 15, 21, 8, 10, 25, 27, 63, 12, 11, 35, 33, 81, 147, 16, 13, 45, 39, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113
OFFSET
1,2
COMMENTS
This is a permutation of the positive integers.
All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018
Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022
EXAMPLE
Array begins:
1, 3, 9, 21, 63, 147, 357, 903, 2499, 6069, ...
2, 5, 15, 27, 81, 171, 399, 987, 2709, 6321, ...
4, 7, 25, 33, 99, 189, 441, 1029, 2793, 6325, ...
6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
...
[Lower right hand triangle of array completed by Hartmut F. W. Hoft, Oct 04 2022]
MATHEMATICA
(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]<r, AppendTo[list[[parts]], k]]]; Transpose[PadRight[list, {c, r}, "..."]]]
a240062[n_, r_] := Module[{arr=widthTable[n, {r, r}], vec=Table[0, PolygonalNumber[r]], i, j}, For[i=1, i<=r, i++, For[j=r-i+1, j>=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 06 2014
EXTENSIONS
a(n) > 128 from Michel Marcus, Apr 08 2014
STATUS
approved