This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357581 #16 Oct 11 2022 01:01:57
%S A357581 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 A357581 1250,171,729,128,14,121,51,2401,207,15625,903,256,17,169,55,4802,243,
%U A357581 31250,987,3025,512,19,242,57,14641,261,117649,1029,3249,6875
%N A357581 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 A357581 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 A357581 This sequence is a subsequence of A240062 and each column in this sequence is a subsequence in the respective column of A240062.
%e A357581 The upper left hand 11 X 11 section of the table for a(n) <= 2*10^7:
%e A357581 1 2 3 4 5 6 7 8 9 10 11 ...
%e A357581 ----------------------------------------------------------------------
%e A357581 1 3 9 21 81 147 729 903 3025 6875 59049
%e A357581 2 5 25 27 625 171 15625 987 3249 7203 9765625
%e A357581 4 7 49 33 1250 207 31250 1029 4761 13203 19531250
%e A357581 8 10 50 39 2401 243 117649 1113 6561 13527 ...
%e A357581 16 11 98 51 4802 261 235298 1239 7569 14013 ...
%e A357581 32 13 121 55 14641 275 1771561 1265 8649 14499 ...
%e A357581 64 14 169 57 28561 279 3543122 1281 12321 14661 ...
%e A357581 128 17 242 65 29282 333 4826809 1375 14161 15471 ...
%e A357581 256 19 289 69 57122 363 7086244 1407 15129 15633 ...
%e A357581 512 22 338 85 58564 369 9653618 1491 16641 15957 ...
%e A357581 1024 23 361 87 83521 387 19307236 1533 17689 16119 ...
%e A357581 ...
%e A357581 Each column k > 1 contains odd and even numbers since, e.g., 5^(k-1) and 2 * 5^(k-1) belong to it.
%e A357581 Column 1: A000079, subsequence of A174973 = A238443, and of column 1 in A240062.
%e A357581 Column 2: A246955, subsequence of A239929; 78 is the smallest number not in A246955.
%e A357581 Column 3: A247687, subsequence of A279102; 15 is the smallest number not in A247687.
%e A357581 Odd numbers in column 3: A001248(k), k > 1.
%e A357581 Column 4: A264102, subsequence of A280107; 75 is the smallest number not in A264102.
%e A357581 Column 5: subsequence of A320066; 63 = A320066(1) is not in column 5.
%e A357581 Numbers in column 5 have the form 2^k * p^4 with p > 2 prime and 0 <= k < floor(log_2(p)).
%e A357581 Odd numbers in column 5: A030514(k), k > 1.
%e A357581 Column 6: subsequence of A320511; 189 is the smallest number not in column 6.
%e A357581 Smallest even number in column 6 is 5050.
%e A357581 Column 7: Numbers have the form 2^k * p^6 with p > 2 prime and 0 <= k < floor(log_2(p)).
%e A357581 Odd numbers in column 7: A030516(k), k > 1.
%e A357581 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 A357581 (* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
%t A357581 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 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 A357581 a357581T[n_, r_] := TableForm[width1Table[n, {r, r}]]
%t A357581 a357581[120000, 10] (* sequence data - first 10 antidiagonals *)
%t A357581 a357581T[120000, 10] (* upper left hand 10x10 array *)
%t A357581 a357581T[20000000, 11] (* 11x11 array - very long computation time *)
%Y A357581 Cf. A000079, A001248, A030514, A030516, A174905, A174973, A237593, A238443, A239929, A241008, A241010, A246955, A247687, A264102, A279102, A280107, A318843, A320066, A320511, A341969, A341970, A341971.
%K A357581 nonn,tabl
%O A357581 1,2
%A A357581 _Hartmut F. W. Hoft_, Oct 04 2022