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A360461 T(n,k) is the sum of all the k-th smallest divisors of all positive integers <= n. Irregular triangle read by rows (n>=1, k>=1).

%I A360461 #59 Mar 06 2023 07:51:03
%S A360461 1,2,2,3,5,4,7,4,5,12,4,6,14,7,6,7,21,7,6,8,23,11,14,9,26,20,14,10,28,
%T A360461 25,24,11,39,25,24,12,41,28,28,6,12,13,54,28,28,6,12,14,56,35,42,6,12,
%U A360461 15,59,40,57,6,12,16,61,44,65,22,12,17,78,44,65,22,12,18,80,47,71,31,30,19,99,47,71,31,30
%N A360461 T(n,k) is the sum of all the k-th smallest divisors of all positive integers <= n. Irregular triangle read by rows (n>=1, k>=1).
%C A360461 Also, looking at all the partitions into equal-sized parts of all positive integers <= n, T(n,k) is the total number of parts in the partitions with the k-th largest parts.
%C A360461 Column k lists the partial sums of the column k of A027750.
%C A360461 The rows where the length row increases to a record gives A002182.
%H A360461 Paolo Xausa, <a href="/A360461/b360461.txt">Table of n, a(n) for n = 1..10175 (rows 1..550 of triangle, flattened)</a>
%H A360461 Michael De Vlieger, <a href="/A360461/a360461.png">Log log scatterplot of row n</a>, n = 1..5040, showing T(n, k) with trajectory of k in a color function where black indicates k = 1, red k = 2, ..., magenta k = 60, ignoring zeros.
%e A360461 Triangle begins:
%e A360461    1;
%e A360461    2,   2;
%e A360461    3,   5;
%e A360461    4,   7,  4;
%e A360461    5,  12,  4;
%e A360461    6,  14,  7,   6;
%e A360461    7,  21,  7,   6;
%e A360461    8,  23, 11,  14;
%e A360461    9,  26, 20,  14;
%e A360461   10,  28, 25,  24;
%e A360461   11,  39, 25,  24;
%e A360461   12,  41, 28,  28,  6, 12;
%e A360461   ...
%e A360461 For n = 6 the divisors, in increasing order, of all positive integers <= 6 are as follows:
%e A360461   -----------------------------
%e A360461   n\k |    1     2     3     4
%e A360461   -----------------------------
%e A360461   1   |    1
%e A360461   2   |    1,    2
%e A360461   3   |    1,    3
%e A360461   4   |    1,    2,    4
%e A360461   5   |    1,    5
%e A360461   6   |    1,    2,    3,    6
%e A360461 .
%e A360461 The sum of the first divisors (k = 1) is equal to 1+1+1+1+1+1 = 6, so T(6,1) = 6.
%e A360461 The sum of the second divisors (k = 2) is equal to 2+3+2+5+2 = 14, so T(6,2) = 14.
%e A360461 The sum of the third divisors (k = 3) is equal to 4+3 = 7, so T(6,3) = 7.
%e A360461 The sum of the fourth divisors (k = 4) is equal to 6, so T(6,4) = 6.
%e A360461 So the 6th row of the triangle is [6, 14, 7, 6].
%e A360461 Also, for n = 6 the partitions into equal parts, with the sizes of the parts in decreasing order, of all positive integers <= 6 are as follows:
%e A360461   ----------------------------------------------------
%e A360461   n\k |     1      2              3           4
%e A360461   ----------------------------------------------------
%e A360461   1   |    [1]
%e A360461   2   |    [2],  [1,1]
%e A360461   3   |    [3],  [1,1,1]
%e A360461   4   |    [4],  [2,2],       [1,1,1,1]
%e A360461   5   |    [5],  [1,1,1,1,1]
%e A360461   6   |    [6],  [3,3],       [2,2,2],   [1,1,1,1,1,1]
%e A360461 .
%e A360461 The total number of parts in the 1st partitions (k = 1) is 6, so T(6,1) = 6.
%e A360461 The total number of parts in the 2nd partitions (k = 2) is 14, so T(6,2) = 14.
%e A360461 The total number of parts in the 3rd partitions (k = 3) is 7, so T(6,3) = 7.
%e A360461 The total number of parts in the 4th partitions (k = 4) is 6, so T(6,4) = 6.
%e A360461 So the 6th row of the triangle is [6, 14, 7, 6].
%t A360461 nn = 20; s[_] = 0; m[0] = 0; Do[Set[m[n], Max[m[n - 1], DivisorSigma[0, n]]], {n, nn}]; Do[MapIndexed[(s[First[#2]] += #1; Set[t[n, First[#2]], s[First[#2]]]) &, PadRight[Divisors[n], m[n]]], {n, nn}]; Table[t[n, k], {n, nn}, {k, m[n]}] // Flatten (* _Michael De Vlieger_, Mar 04 2023 *)
%t A360461 A360461[rowmax_]:=DeleteCases[Accumulate[PadRight[Divisors[Range[rowmax]]]],0,{2}];
%t A360461 A360461[20] (* Generates 20 rows *) (* _Paolo Xausa_, Mar 05 2023 *)
%o A360461 (PARI) rowlen(n) = vecmax(vector(n, k, numdiv(k))); \\ A070319
%o A360461 row(n) = my(vd=vector(n, i, divisors(i)), nb=rowlen(n)); vector(nb, k, sum(i=1, #vd, if (#(vd[i]) >= k, vd[i][k]))); \\ _Michel Marcus_, Mar 06 2023
%Y A360461 Row sums give A024916.
%Y A360461 Row lengths give A070319.
%Y A360461 Column 1 gives A000027.
%Y A360461 Column 2 gives A088821.
%Y A360461 The sum of the first n rows gives A175254.
%Y A360461 Main sequences: A027750 and A244051.
%Y A360461 Cf. A000005, A002182.
%K A360461 nonn,tabf,look
%O A360461 1,2
%A A360461 _Omar E. Pol_, Feb 07 2023