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.

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

Author

Amiram Eldar, Mar 29 2019

Keywords

• nonn

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

Cf. A005153, A307223, A307224.

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