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

1,6

LINKS

Table of n, a(n) for n=1..36.

FORMULA

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.

EXAMPLE

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.

MATHEMATICA

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]

CROSSREFS

Cf. A000043, A000396, A005153, A307223.

Sequence in context: A191419 A054373 A061847 * A309595 A329074 A296434

Adjacent sequences:  A307221 A307222 A307223 * A307225 A307226 A307227

KEYWORD

nonn

AUTHOR

Amiram Eldar, Mar 29 2019

STATUS

approved

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Last modified February 18 21:20 EST 2020. Contains 332028 sequences. (Running on oeis4.)