A328368 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.

1, 2, 3, 2, 4, 2, 5, 4, 6, 4, 3, 7, 6, 3, 8, 6, 3, 9, 8, 6, 10, 8, 6, 4, 11, 10, 6, 4, 12, 10, 9, 4, 13, 12, 9, 4, 14, 12, 9, 8, 15, 14, 12, 8, 5, 16, 14, 12, 8, 5, 17, 16, 12, 8, 5, 18, 16, 15, 12, 5, 19, 18, 15, 12, 5, 20, 18, 15, 12, 10, 21, 20, 18, 12, 10, 6, 22, 20, 18, 16, 10, 6, 23, 22, 18, 16, 10, 6

Offset

1,2

Comments

Column k lists k times every nonzero multiple of k in nondecreasing order.

Column k lists the partial sums of the k-th column of triangle A285914.

Links

Table of n, a(n) for n=1..88.

Example

Triangle begins:
   1;
   2;
   3,  2;
   4,  2;
   5,  4;
   6,  4,  3;
   7,  6,  3;
   8,  6,  3;
   9,  8,  6;
  10,  8,  6,  4;
  11, 10,  6,  4;
  12, 10,  9,  4;
  13, 12,  9,  4;
  14, 12,  9,  8;
  15, 14, 12,  8,  5;
  16, 14, 12,  8,  5;
  17, 16, 12,  8,  5;
  18, 16, 15, 12,  5;
  19, 18, 15, 12,  5;
  20, 18, 15, 12, 10;
  21, 20, 18, 12, 10,  6;
  22, 20, 18, 16, 10,  6;
  23, 22, 18, 16, 10,  6;
  24, 22, 21, 16, 10,  6;
  25, 24, 21, 16, 15,  6;
  26, 24, 21, 20, 15,  6;
  27, 26, 24, 20, 15, 12;
  28, 26, 24, 20, 15, 12, 7;
...

Prog

(PARI) tt(n, k) = k*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

Crossrefs

Row sums give A285899.

Row n has length A003056(n).

Column 1 gives A000027.

Column k starts with k in the row A000217(k).

Cf. A052928, A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.

Sequence in context: A222817 A344324 A309428 * A052297 A365049 A297162

Adjacent sequences: A328365 A328366 A328367 * A328369 A328370 A328371

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 02 2019

Status

approved