A143932 -id:A143932

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A143933

a(n) is the smallest prime x such that x^2-n! is also prime.

+10
3

2, 2, 3, 11, 19, 31, 79, 211, 607, 1931, 6337, 21961, 78919, 295291, 1143563, 4574149, 18859777, 80014843, 348776611, 1559776279, 7147792903, 33526120129, 160785623729, 787685471519, 3938427356629, 20082117944579, 104349745809137, 552166953567737

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Every prime > 3 in this sequence is bigger than the n-th prime, see comment to A121926. For the smallest number x such that x^2-n! is prime see A143931. For the smallest prime numbers of the form x^2-n! see A143932.

LINKS

Robert Israel, Table of n, a(n) for n = 1..300

MAPLE

f:= proc(n) local p, t;

t:= n!;

p:= floor(sqrt(t));

p:= nextprime(p);

if isprime(p^2-t) then return p fi

end proc:

map(f, [$1..28]); # Robert Israel, Feb 10 2019

MATHEMATICA

f[n_] := Block[{p = NextPrime[ Sqrt[ n!]]}, While[ !PrimeQ[p^2 - n!], p = NextPrime@ p]; p]; Array[f, 27] (* Robert G. Wilson v, Jan 08 2015 *)

PROG

(PARI) a(n)=my(N=n!, x=sqrtint(N)+1); while(!isprime(x^2-N), x=nextprime(x+1)); x \\ Charles R Greathouse IV, Dec 09 2014

CROSSREFS

Cf. A121926, A143931, A143932.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Sep 05 2008

EXTENSIONS

Corrected by Charles R Greathouse IV, Dec 09 2014

STATUS

approved

A143931

a(n) is the smallest positive integer x such that x^2 - n! is prime.

+10
2

2, 2, 3, 11, 19, 31, 79, 209, 607, 1921, 6337, 21907, 78913, 295289, 1143539, 4574149, 18859733, 80014841, 348776611, 1559776279, 7147792823, 33526120127, 160785623627, 787685471389, 3938427356623, 20082117944263, 104349745809077

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For the smallest positive prime numbers of the form x^2 - n! see A143932.

For primes x in this sequence see A143933.

LINKS

Table of n, a(n) for n=1..27.

EXAMPLE

a(1)=2 because 2^2-1! = 3 is prime;

a(2)=2 because 2^2-2! = 2 is prime;

a(3)=3 because 3^2-3! = 3 is prime;

a(4)=11 because 11^2-4! = 97 is prime.

MATHEMATICA

a = {}; Do[k = Round[Sqrt[n! ]] + 1; While[ ! PrimeQ[k^2 - n! ], k++ ]; AppendTo[a, k], {n, 1, 50}]; a

spi[n_]:=Module[{k=Ceiling[Sqrt[n!]], nf=n!}, While[!PrimeQ[k^2-nf], k++]; k]; Array[ spi, 30] (* Harvey P. Dale, Feb 17 2023 *)