A386755 -id:A386755 - cp's OEIS frontend

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

Tamas Sandor Nagy, Jul 24 2025

nonn

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

			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.

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

Cf. A386755.

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

\\ 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

\\ 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

A386949 Irregular triangle whose n-th row lists the nonzero terms of the n-th column of A386755.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 3, 3, 1, 2, 2, 4, 4, 1, 5, 5, 5, 5, 5, 5, 1, 2, 2, 3, 3, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 7, 7, 1, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 1, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 9, 9, 1, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 10, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Michel Marcus, Aug 10 2025

			Triangle begins:
  1;
  1, 2, 2;
  1, 3, 3, 3, 3;
  1, 2, 2, 4, 4;
  1, 5, 5, 5, 5, 5, 5;
  1, 2, 2, 3, 3, 6, 6, 6, 6;
  1, 7, 7, 7, 7, 7, 7, 7, 7;
  1, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8;
  1, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 9, 9;
  1, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 10, 10;
  ...

Cf. A386755 (original triangle), A386520 (row sums).

Cf. A027750.

orow(n) = my(v=vector(n), m=n); for(k=1, n, my(keepm = m); while(m%k, m--); if (m == 0, keepm=m, v[m] = k; m--); ); v; \\ A386755
nrow(n) = my(ok=1, k=1, last=-1, list=List(), r); while(ok, r=row(k); if ((#r >= n) && r[n], listput(list, r[n])); k++; if (#r>=n, if ((last==n) && (r[n]==0), ok = 0, last = r[n]))); Vec(list);

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A386520 Column sums of the triangle in A386755.

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A386949 Irregular triangle whose n-th row lists the nonzero terms of the n-th column of A386755.

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PARI