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Search: id:A111392
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| A111392 |
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a(n) = Product_{i=1..n-1} (Product_{k=1..i} p_k + Product_{k=i+1..n} p_k). |
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+0
5
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| 2, 5, 187, 162319, 10697595389, 63619487169453143, 74365399061678006800073593, 11864736003419293844093922527852416537, 601642845102734414280661105098046392912578705726003
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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This sequence gives another proof that there are infinitely many primes. Let N = Product_{1<=i<n} (Product_{1<=k<=i} p_k + Product_{i<k<=n} p_k). Suppose there are only a finite number of primes p_i, 1<=i<=n. If N is prime, then for all i, not(N=p_i). Because, for all i, p_i<N. If N is composite, then it must have a prime divisor p which is different from primes p_i. Because, for all i, not(N==0 mod p_i).
a(1) could also be chosen to be 1.
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MATHEMATICA
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f[n_] := Product[(Product[Prime[k], {k, i}] + Product[Prime[k], {k, i + 1, n}]), {i, n - 1}]; Array[f, 9] (* Robert G. Wilson v *)
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PROGRAM
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t=10; for(n=2, t, print(prod(i=1, n-1, prod(k=1, i, prime(k))+prod(k=i+1, n, prime(k))))); (Gerald McGarvey)
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CROSSREFS
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Cf. A024451, A112404.
Sequence in context: A041505 A124275 A013130 this_sequence A100366 A012975 A012954
Adjacent sequences: A111389 A111390 A111391 this_sequence A111393 A111394 A111395
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KEYWORD
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nonn,nice
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AUTHOR
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Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Nov 08 2005
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EXTENSIONS
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Corrected and extended by Gerald McGarvey (Gerald.McGarvey(AT)comcast.net) and Robert G. Wilson v (rgwv(at)rgwv.com), Nov 12 2005
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