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A318670
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Number of subsets of divisors of n whose least common multiple is n and the sum does not exceed n. For n > 1, 1 is excluded from the set of divisors.
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4
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 6, 1, 6, 2, 2, 1, 17, 1, 2, 1, 7, 1, 34, 1, 1, 2, 2, 2, 44, 1, 2, 2, 23, 1, 36, 1, 7, 7, 2, 1, 65, 1, 7, 2, 7, 1, 21, 2, 25, 2, 2, 1, 471, 1, 2, 7, 1, 2, 39, 1, 7, 2, 44, 1, 355, 1, 2, 7, 7, 2, 39, 1, 89, 1, 2, 1, 531, 2, 2, 2, 27, 1, 559, 2, 7, 2, 2, 2, 257, 1, 7, 7, 61, 1, 39, 1, 28, 46
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OFFSET
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1,6
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COMMENTS
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These count the "starter sets" employed by a simple backtracking algorithm that computes A317624. See the PARI program dated Sep 08-10 2018 under that entry.
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LINKS
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FORMULA
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For all n >= 1:
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EXAMPLE
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For n = 45, there exists the following subsets of its divisors larger than one (3, 5, 9, 15, 45) that satisfy the condition that the least common multiple of the members is 45, and the sum does not exceed 45: (45), (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15) and (5, 9, 15), altogether seven subsets, thus a(45) = 7.
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PROG
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(PARI)
A318670(n) = if(1==n, 1, my(ds=(divisors(n)[2..numdiv(n)]), subsets = select(v -> ((vecsum(v)<=n)&&(n==lcm(v))), powerset(ds))); length(subsets)); \\ A memory-hog implementation.
powerset(v) = { my(siz=2^length(v), pv=vector(siz)); for(i=0, siz-1, pv[i+1] = choosebybits(v, i)); pv; };
choosebybits(v, m) = { my(s=vector(hammingweight(m)), i=j=1); while(m>0, if(m%2, s[j] = v[i]; j++); i++; m >>= 1); s; }; \\ Antti Karttunen, Sep 08 2018
(PARI)
\\ A better program:
strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
A318670aux(orgn, n, parts, from=1, ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn, s++); for(i=from, k, if(parts[i]<=n, newss = List(ss); listput(newss, parts[i]); s += A318670aux(orgn, n-parts[i], parts, i+1, newss))); (s) };
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CROSSREFS
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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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