%I #16 Oct 11 2022 01:01:57

%S 1,2,3,4,5,9,8,7,25,21,16,10,49,27,81,32,11,50,33,625,147,64,13,98,39,

%T 1250,171,729,128,14,121,51,2401,207,15625,903,256,17,169,55,4802,243,

%U 31250,987,3025,512,19,242,57,14641,261,117649,1029,3249,6875

%N Square array read by antidiagonals of numbers whose symmetric representation of sigma consists only of parts that have width 1; column k indicates the number of parts and row n indicates the n-th number in increasing order in each of the columns.

%C This sequence is a permutation of A174905. Numbers in the even numbered columns of the table form A241008 and those in the odd numbered columns form A241010. The first row of the table is A318843.

%C This sequence is a subsequence of A240062 and each column in this sequence is a subsequence in the respective column of A240062.

%e The upper left hand 11 X 11 section of the table for a(n) <= 2*10^7:

%e 1 2 3 4 5 6 7 8 9 10 11 ...

%e ----------------------------------------------------------------------

%e 1 3 9 21 81 147 729 903 3025 6875 59049

%e 2 5 25 27 625 171 15625 987 3249 7203 9765625

%e 4 7 49 33 1250 207 31250 1029 4761 13203 19531250

%e 8 10 50 39 2401 243 117649 1113 6561 13527 ...

%e 16 11 98 51 4802 261 235298 1239 7569 14013 ...

%e 32 13 121 55 14641 275 1771561 1265 8649 14499 ...

%e 64 14 169 57 28561 279 3543122 1281 12321 14661 ...

%e 128 17 242 65 29282 333 4826809 1375 14161 15471 ...

%e 256 19 289 69 57122 363 7086244 1407 15129 15633 ...

%e 512 22 338 85 58564 369 9653618 1491 16641 15957 ...

%e 1024 23 361 87 83521 387 19307236 1533 17689 16119 ...

%e ...

%e Each column k > 1 contains odd and even numbers since, e.g., 5^(k-1) and 2 * 5^(k-1) belong to it.

%e Column 1: A000079, subsequence of A174973 = A238443, and of column 1 in A240062.

%e Column 2: A246955, subsequence of A239929; 78 is the smallest number not in A246955.

%e Column 3: A247687, subsequence of A279102; 15 is the smallest number not in A247687.

%e Odd numbers in column 3: A001248(k), k > 1.

%e Column 4: A264102, subsequence of A280107; 75 is the smallest number not in A264102.

%e Column 5: subsequence of A320066; 63 = A320066(1) is not in column 5.

%e Numbers in column 5 have the form 2^k * p^4 with p > 2 prime and 0 <= k < floor(log_2(p)).

%e Odd numbers in column 5: A030514(k), k > 1.

%e Column 6: subsequence of A320511; 189 is the smallest number not in column 6.

%e Smallest even number in column 6 is 5050.

%e Column 7: Numbers have the form 2^k * p^6 with p > 2 prime and 0 <= k < floor(log_2(p)).

%e Odd numbers in column 7: A030516(k), k > 1.

%e Numbers in the column numbered with the n-th prime p_n have the form: 2^k * p^(p_n - 1) with p > 2 prime and 0 <= k < floor(log_2(p_n)).

%t (* function a341969 and support functions are defined in A341969, A341970 and A341971 *)

%t width1Table[n_, {r_, c_}] := Module[{k, list=Table[{}, c], wL, wLen, pCount, colLen}, For[k=1, k<=n, k++, wL=a341969[k]; wLen=Length[wL]; pCount=(wLen+1)/2; If[pCount<=c&&Length[list[[pCount]]]<r&&SubsetQ[{0, 1}, Union[wL]], AppendTo[list[[pCount]], k]]]; Transpose[PadRight[list, {c, r},"..."]]]

%t a357581[n_, r_] := Module[{arr=width1Table[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]

%t a357581T[n_, r_] := TableForm[width1Table[n, {r, r}]]

%t a357581[120000, 10] (* sequence data - first 10 antidiagonals *)

%t a357581T[120000, 10] (* upper left hand 10x10 array *)

%t a357581T[20000000, 11] (* 11x11 array - very long computation time *)

%Y Cf. A000079, A001248, A030514, A030516, A174905, A174973, A237593, A238443, A239929, A241008, A241010, A246955, A247687, A264102, A279102, A280107, A318843, A320066, A320511, A341969, A341970, A341971.

%K nonn,tabl

%O 1,2

%A _Hartmut F. W. Hoft_, Oct 04 2022