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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
OFFSET
1,6
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 29 2019
STATUS
approved