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A191150
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Hypersigma(n): sum of the divisors of n plus the recursive sum of the divisors of the restricted divisors.
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4
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1, 3, 4, 10, 6, 19, 8, 28, 17, 27, 12, 64, 14, 35, 34, 72, 18, 82, 20, 88, 44, 51, 24, 188, 37, 59, 61, 112, 30, 165, 32, 176, 64, 75, 62, 290, 38, 83, 74, 252, 42, 209, 44, 160, 139, 99, 48, 512, 65, 166, 94, 184, 54, 306, 90, 316, 104, 123, 60, 588, 62, 131
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OFFSET
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1,2
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COMMENTS
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First we add up all the divisors of n, and then we add in the divisors of each restricted divisor of n (not 1 or n itself) and continue the recursion until such a depth as that there only numbers with no restricted divisors (prime numbers).
Thus if n is prime then hypersigma(n) is the same as sigma(n).
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LINKS
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FORMULA
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EXAMPLE
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a(12) = 64 since: the sum of the divisors of 12 is 28; to 28 we add 3 and 4 (corresponding to the prime divisors 2 and 3) bringing us up to 35; for 4 and 6 we continue the recursion, with 4 bringing us up to 45 and 6 brings up to 64.
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MAPLE
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a:= proc(n) option remember; uses numtheory;
sigma(n)+add(a(d), d=divisors(n) minus {1, n})
end:
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MATHEMATICA
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hyperSigma[1] := 1; hyperSigma[n_] := hyperSigma[n] = Module[{d=Divisors[n]}, Total[d] + Total[hyperSigma /@ Rest[Most[d]]]]; Table[hyperSigma[n], {n, 100}] (* From T. D. Noe with a slight modification *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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