A307223 -id:A307223 - cp's OEIS frontend

A307224 The total number of combinations for presenting the set of numbers 1 <= k <= sigma(n) as sums of distinct divisors of n.

1, 1, 0, 1, 0, 8, 0, 1, 0, 0, 0, 1088391168, 0, 0, 0, 1, 0, 2985984, 0, 2097152, 0, 0, 0, 103312130400000000000000000000000000, 0, 0, 0, 128, 0, 5888655348399321787662336000000000000, 0, 1, 0, 0, 0, 1373825949385418214640573033104853375673916456960000000000000000

Offset: 1

Amiram Eldar, Mar 29 2019

nonn

			a(6) = 8 since the divisors of 6 are {1, 2, 3, 6}, k = 3, 6, and 9 are each the sum of 2 subsets (3: {1,2} and {3}, 6: {1,2,3} and {6}, 9: {1,2,6} and {3,6}) and the other values of k are sums of a single subset. Thus, a(6) = 1*1*2*1*1*2*1*1*2*1*1*1 = 8.

Cf. A000043, A000396, A005153, A307223.

T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; a[n_] := Product[T[n, k], {k,1,DivisorSigma[1,n]}]; Array[a, 50]

Equals Product_{k=1..sigma(n)} T(n, k), where T(n, k) is given in A307223.
a(n) > 0 iff n is practical (A005153).
a(2^k) = 1.
a(2^(p-1)*(2^p-1)) = 2^(2^p-1) for Mersenne exponents p.

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

Amiram Eldar, Mar 29 2019

nonn

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, ...

Cf. A005153, A307223, A307224.

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

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

More terms from David A. Corneth, Mar 29 2019

cp's OEIS Frontend

A307224 The total number of combinations for presenting the set of numbers 1 <= k <= sigma(n) as sums of distinct divisors of n.

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