A386520 Column sums of the triangle in A386755.

1, 5, 13, 13, 31, 35, 57, 61, 85, 85, 111, 99, 235, 89, 353, 173, 171, 341, 343, 229, 489, 423, 415, 435, 661, 525, 535, 559, 1161, 427, 931, 653, 1201, 787, 941, 885, 1629, 537, 1443, 1839, 1723, 931, 1119, 1525, 2415, 741, 2257, 2327, 1947, 2005, 2767, 1131, 3181, 1055, 3131, 2147

Offset

1,2

Comments

It appears that A007952(n) is the index of the row where n first appears.

It appears that A007952(n)-1 is the index of the row where the last nonzero term of the n-th column is seen. - Michel Marcus, Aug 02 2025

Links

Michel Marcus, Table of n, a(n) for n = 1..500

Example

Triangle whose columns are summed.
  m/n| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  ----------------------------------------------------------------
   1 | 1
   2 |    1
   3 |    2  1
   4 |    2     1
   5 |       3  2  1
   6 |       3  2     1
   7 |       3        2  1
   8 |       3        2     1
   9 |          4     3     2  1
  10 |          4     3     2     1
  11 |             5        4  3  2  1
  12 |             5        4  3  2     1
  13 |             5        4  3        2  1
  14 |             5        4  3        2     1
  15 |             5        4           3     2  1
  16 |             5        4           3     2     1
  17 |                6           5     4        3  2  1
  18 |                6           5     4        3  2     1
  19 |                6           5     4        3        2  1
  20 |                6           5     4        3        2     1
  ...
The completed column for n=5 is definitely fully visible here because in column 6 for n=6 the divisor k=6 already appeared. That means that column 5 cannot have more divisors in it under the last k=5 in row 17 because in that row only k=7 may follow k=6 in theory, but 7 does not divide 5. So, all similarly proven, definitely fully visible completed columns in this sample array are readily summable by sight. E.g. column 5: a(5) = 1 + 5 + 5 + 5 + 5 + 5 + 5 = 31.

Prog

(PARI) \\ uses row(n) from A386755
a(n) = my(ok=1, k=1, last=-1, s=0, r); while(ok, r=row(k); if (#r >= n, s+=r[n]); k++; if (#r>=n, if ((last==n) && (r[n]==0), ok = 0, last = r[n]))); s; \\ Michel Marcus, Aug 02 2025
(PARI) \\ uses row(n) from A386755
lista(nn) = my(ok=1, k=1, vlast=vector(nn, i, -1), vs=vector(nn)); while(ok, my(r=row(k)); for (i=1, nn, if (#r>=i, vs[i]+=r[i])); k++; my(nbok=0); for (i=1, nn, if (#r>=i, if ((vlast[i]==i) && (r[i]==0), nbok++, vlast[i] = r[i]))); if (nbok == nn, ok = 0); ); vs; \\ Michel Marcus, Aug 02 2025

Crossrefs

Cf. A386755.

Cf. A007952 (row number where k=n first appears).

Sequence in context: A232610 A235337 A151994 * A321992 A231806 A183782

Adjacent sequences: A386517 A386518 A386519 * A386521 A386522 A386523

Keyword

nonn

Author

Tamas Sandor Nagy, Jul 24 2025

Status

approved