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A085493
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Numbers k having partitions into distinct divisors of k + 1.
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7
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1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 47, 53, 55, 59, 63, 65, 69, 71, 77, 79, 83, 87, 89, 95, 99, 103, 107, 111, 119, 125, 127, 131, 139, 143, 149, 155, 159, 161, 167, 175, 179, 191, 195, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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{k > 0 : 0 < [x^k] Product_{d divides (k+1)} (1+x^d)}. - Alois P. Heinz, Feb 04 2023
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EXAMPLE
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The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. Since 6 + 14 + 21 = 41, 41 is in the sequence.
The divisors of 43 are 1, 43. Since no selection of these divisors can possibly add up to 42, this means that 42 is not in the sequence.
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MAPLE
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q:= proc(m) option remember; local b, l; b, l:=
proc(n, i) option remember; n=0 or i>=1 and
(l[i]<=n and b(n-l[i], i-1) or b(n, i-1))
end, sort([numtheory[divisors](m+1)[]]);
b(m, nops(l)-1)
end:
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MATHEMATICA
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divNextableQ[n_] := TrueQ[Length[Select[Subsets[Divisors[n + 1]], Plus@@# == n &]] > 0]; Select[Range[100], divNextableQ] (* Alonso del Arte, Jan 07 2023 *)
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PROG
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(Scala) def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
def divPartSums(n: Int): List[Int] = divisors(n).toSet.subsets.toList.map(_.sum)
(1 to 128).filter(n => divPartSums(n + 1).contains(n)) // Alonso del Arte, Jan 26 2023
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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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