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A305611
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Number of distinct positive subset-sums of the multiset of prime factors of n.
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11
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0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 9, 1, 3, 5, 6, 3, 7, 1, 5, 3, 6, 1, 10, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 10, 3, 3
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OFFSET
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1,4
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COMMENTS
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An integer n is a positive subset-sum of a multiset y if there exists a nonempty submultiset of y with sum n.
One less than the number of distinct values obtained when A001414 is applied to all divisors of n. - Antti Karttunen, Jun 13 2018
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LINKS
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EXAMPLE
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The a(12) = 5 positive subset-sums of {2, 2, 3} are 2, 3, 4, 5, and 7.
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MATHEMATICA
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Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n], {p_, k_}:>Table[p, {k}]]]]]], {n, 100}]
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PROG
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(PARI)
up_to = 65537;
v001414 = vector(up_to, n, A001414(n));
A305611(n) = { my(m=Map(), s, k=0); fordiv(n, d, if(!mapisdefined(m, s = v001414[d]), mapput(m, s, s); k++)); (k-1); }; \\ Antti Karttunen, Jun 13 2018
(Python)
from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
fs = factorint(n)
return len(set(sum(d) for i in range(1, sum(fs.values())+1) for d in multiset_combinations(fs, i))) # Chai Wah Wu, Aug 23 2021
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CROSSREFS
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Cf. A001414, A027746, A056239, A122768, A276024, A284640, A299701, A299702, A301855, A301935, A301957, A304792, A304793, A305613.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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