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A090975 Least integer k such that n!+1-k is prime. 0
0, 0, 0, 0, 2, 8, 2, 2, 32, 14, 12, 0, 2, 24, 2, 48, 54, 60, 42, 102, 32, 32, 74, 90, 74, 150, 38, 0, 102, 32, 2, 62, 2, 2, 194, 114, 128, 0, 2, 74, 84, 0, 80, 110, 110, 54, 90, 80, 104, 60, 98, 180, 68, 60, 128, 62, 462, 278, 110, 138, 140, 72, 72, 102, 360, 128, 318, 192 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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. n!+1 itself may be a prime number as in the case of n=3, 11, 27 (see A002981 for all such n). Now a(n) measures, when the first prime number previous to n!+2 appears. Thus a(n)=8 means that n!+1-3 is prime and so on. Obviously, the values of a(n) are always even numbers. Conjectures: |a(n)-1| is either 1 or a prime number. Is the growth of b(n) := sum(a(k),k=0..n) quadratic, that is b(n)=O(n^2)?

LINKS

Table of n, a(n) for n=0..67.

EXAMPLE

a(3)=0 because 3!+1-0=7 is prime.

a(4)=2 because 4!+1-2=23 is prime and 24 and 25 are not.

MAPLE

a := proc(n) option remember; n!+1-prevprime(n!+2); end;

CROSSREFS

Cf. A090786, A002981, A005095.

Sequence in context: A076344 A307450 A173686 * A257579 A199715 A296049

Adjacent sequences:  A090972 A090973 A090974 * A090976 A090977 A090978

KEYWORD

nonn

AUTHOR

Frederick Magata (frederick.magata(AT)uni-muenster.de), Feb 28 2004

STATUS

approved

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Last modified August 5 19:31 EDT 2021. Contains 346488 sequences. (Running on oeis4.)