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A109361
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a(n) = Product_{k=1..n} sigma(k)/d(k), where sigma(k) = Sum_{j|k} j and d(k) = Sum_{j|k} 1. Set a(n) = 0 if the corresponding product is not an integer (e.g., for n=2 and n=10).
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1
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1, 0, 3, 7, 21, 63, 252, 945, 4095, 0, 110565, 515970, 3611790, 21670740, 130024440, 806151528, 7255363752, 47159864388, 471598643880, 3301190507160, 26409524057280, 237685716515520, 2852228598186240, 21391714486396800
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OFFSET
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1,3
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COMMENTS
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The product at n = 2 is the noninteger 1.5. The product at n = 10 is the noninteger 18427.5. Jack Brennen's observed that the only values which are not integers occur when n = 2 or 10, for n < 5000. Are all products for n >= 11 integers?
No other nonintegers found up to 200000. - Michel Marcus, Sep 14 2015
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LINKS
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FORMULA
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Product_{k=1..n} sigma(k)/d(k) = Product_{p=primes} Product_{k>=1} ((p^(k+1)-1)*k/((p^k -1)(k+1)))^floor(n/p^k).
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EXAMPLE
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a(4) = 1 * 3 * 4 * 7 /(1 * 2 * 2 * 3) = 7.
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MAPLE
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p:= 1: A[1]:= 1:
for n from 2 to 50 do
p:= p * numtheory:-sigma(n)/numtheory:-tau(n);
if p::integer then A[n]:= p else A[n]:= 0 fi
od:
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MATHEMATICA
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Table[If[IntegerQ[Product[DivisorSigma[1, k]/Length[Divisors[k]], {k, 1, n}]], Product[DivisorSigma[1, k]/Length[Divisors[k]], {k, 1, n}], 0], {n, 1, 30}] (* Stefan Steinerberger, Oct 24 2007 *)
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PROG
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(PARI) a(n) = my(q = prod(k=1, n, sigma(k)/numdiv(k))); if (denominator(q)==1, q, 0); \\ Michel Marcus, Sep 14 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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