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URL: https://oeis.org/A386520

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A386520
Column sums of the triangle in A386755.
3
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
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
KEYWORD
nonn
AUTHOR
Tamas Sandor Nagy, Jul 24 2025
STATUS
approved