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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 17 12:55 EDT 2021. Contains 343971 sequences. (Running on oeis4.)