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

1,2

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, ...

Links

Table of n, a(n) for n=1..53.

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

Prog

(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

Crossrefs

Cf. A005153, A307223, A307224.

Sequence in context: A188158 A061822 A226453 * A261476 A119840 A298033

Adjacent sequences: A307222 A307223 A307224 * A307226 A307227 A307228

Keyword

nonn

Author

Amiram Eldar, Mar 29 2019

Extensions

More terms from David A. Corneth, Mar 29 2019

Status

approved