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A090786
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Least integer k such that n!+n+k+1 is prime.
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2
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0, 0, 0, 1, 0, 1, 0, 3, 14, 7, 0, 5, 16, 53, 4, 27, 6, 13, 18, 69, 8, 9, 8, 73, 106, 15, 32, 19, 38, 193, 76, 95, 46, 3, 62, 25, 94, 273, 4, 57, 12, 19, 54, 27, 2, 193, 54, 185, 4, 33, 10, 219, 0, 17, 168, 15, 92, 49, 224, 233, 210, 707, 68, 207, 2, 127, 216, 5, 14, 61, 68, 785
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| The (n-1) consecutive numbers n!+2,...,n!+n (for n>=2) are not prime. This fact implies that there are arbitrarily large gaps in the distribution of the prime numbers. However, n!+n+1 need not be a prime number. Now a(n) measures, when the next prime number after n!+n appears. Thus a(n)=0 means that n!+n+1 is prime and so on. Obviously, a(n) is parity conserving for n>=2. I.e. if n>=2 then 2 divides n iff 2 divides a(n).
Conjectures: By definition a(n)+n!+1 is prime, but is a(n)+n+1=A037153(n) also a prime number for all n>2? Is the growth of b(n) := sum(a(k),k=0..n) quadratic, that is b(n)=O(n^2)?
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EXAMPLE
| a(5)=1 because 5!+5+1+1=127 is prime and 126 is not.
a(7)=3 because 7!+7+7+1=5051 is prime and 5048, 5049 and 5050 are not prime.
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MAPLE
| a := proc(n) option remember; local r, m, k: r := n!+n: k := 1: m := r+1: while (not isprime(m)) do k := k+1: while (not igcd(k, n)=1) do k := k+1: od: m := r+k: od: k-1; end; or alternatively: a := proc(n) option remember; nextprime(n!+n)-n!-n-1; end;
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CROSSREFS
| Cf. A005095.
Cf. A037153.
Sequence in context: A179087 A058991 A197423 * A127818 A155886 A138959
Adjacent sequences: A090783 A090784 A090785 * A090787 A090788 A090789
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KEYWORD
| nonn
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AUTHOR
| Frederick Magata (fmagata(AT)mi.uni-koeln.de), Feb 09 2004
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