cp's OEIS Frontend

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.

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A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2019

Keywords

Comments

T(n, k) > 0 for all values of k iff n is practical (A005153).

Examples

			Table begins as:
  1
  1,1,1
  1,0,1,1
  1,1,1,1,1,1,1
  1,0,0,0,1,1
  1,1,2,1,1,2,1,1,2,1,1,1
  1,0,0,0,0,0,1,1
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  1,0,1,1,0,0,0,0,1,1,0,1,1
  1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1
		

Crossrefs

Cf. A000203 (row lengths), A307224 (row products).

Programs

  • Mathematica
    T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; Table[T[n, k], {n,1,10}, {k, 1, DivisorSigma[1,n]}] // Flatten

Formula

T(n, n) = A033630(n).
T(n, A030057(n)) = 0 if there is a 0 in the n-th row, i.e. A030057(n) <= sigma(n) or n is not practical.

A307225 Superpractical numbers: practical numbers m with a record total number of combinations for presenting the set of numbers 1 <= k <= sigma(m) as sums of distinct divisors of m.

Original entry on oeis.org

1, 6, 12, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1008, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 6300, 6720, 7560, 9240, 10080, 12600, 13860, 15120, 18480
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2019

Keywords

Comments

Let c(m, k) be the number of ways to present k as the sum of distinct divisors of m, for k=1..sigma(m) (A307223).
Let C(m) = Product_{k=1..sigma(m)} c(m, k) (A307224).
This sequence list (practical) numbers m with a record value of C(m).
The corresponding values of C(m) are 1, 8, 1088391168, 103312130400000000000000000000000000, ...

Crossrefs

Programs

  • Mathematica
    T[n_, k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; f[n_] := Times @@ (T[n, #] & /@ Range[DivisorSigma[1, n]]); s = {}; fmax = 0; Do[f1 = f[n]; If[f1 > fmax, fmax = f1; AppendTo[s, n]], {n, 1, 100}]; s
  • PARI
    upto(n) = {my(v = vector(n, i, print1(i", "); C(i)), r = -1, res = List());
    for(i = 1, n, c = v[i]; if(c > r, listput(res, i); r = c)); res}
    C(n) = {my(v = vector(sigma(n) + 1), t = 1, d = divisors(n)); v[1] = 1; for(i = 1, #d, for(j = 1, t, v[j + d[i]] += v[j] ); t+=d[i] ); vecprod(v) } \\ David A. Corneth, Mar 29 2019

Extensions

More terms from David A. Corneth, Mar 29 2019
Showing 1-2 of 2 results.