login
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.
0
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, ...
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 29 2019
EXTENSIONS
More terms from David A. Corneth, Mar 29 2019
STATUS
approved