This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386520 #88 Aug 11 2025 01:38:52 %S A386520 1,5,13,13,31,35,57,61,85,85,111,99,235,89,353,173,171,341,343,229, %T A386520 489,423,415,435,661,525,535,559,1161,427,931,653,1201,787,941,885, %U A386520 1629,537,1443,1839,1723,931,1119,1525,2415,741,2257,2327,1947,2005,2767,1131,3181,1055,3131,2147 %N A386520 Column sums of the triangle in A386755. %C A386520 It appears that A007952(n) is the index of the row where n first appears. %C A386520 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 %H A386520 Michel Marcus, <a href="/A386520/b386520.txt">Table of n, a(n) for n = 1..500</a> %e A386520 Triangle whose columns are summed. %e A386520 m/n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 %e A386520 ---------------------------------------------------------------- %e A386520 1 | 1 %e A386520 2 | 1 %e A386520 3 | 2 1 %e A386520 4 | 2 1 %e A386520 5 | 3 2 1 %e A386520 6 | 3 2 1 %e A386520 7 | 3 2 1 %e A386520 8 | 3 2 1 %e A386520 9 | 4 3 2 1 %e A386520 10 | 4 3 2 1 %e A386520 11 | 5 4 3 2 1 %e A386520 12 | 5 4 3 2 1 %e A386520 13 | 5 4 3 2 1 %e A386520 14 | 5 4 3 2 1 %e A386520 15 | 5 4 3 2 1 %e A386520 16 | 5 4 3 2 1 %e A386520 17 | 6 5 4 3 2 1 %e A386520 18 | 6 5 4 3 2 1 %e A386520 19 | 6 5 4 3 2 1 %e A386520 20 | 6 5 4 3 2 1 %e A386520 ... %e A386520 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. %o A386520 (PARI) \\ uses row(n) from A386755 %o A386520 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 %o A386520 (PARI) \\ uses row(n) from A386755 %o A386520 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 %Y A386520 Cf. A386755. %Y A386520 Cf. A007952 (row number where k=n first appears). %K A386520 nonn %O A386520 1,2 %A A386520 _Tamas Sandor Nagy_, Jul 24 2025