A124375 -id:A124375 - cp's OEIS frontend

A100614 Numbers n such that (!n)/2 is prime, where !n = Sum_{k=0..n-1} k!.

Original entry on oeis.org

3, 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221, 41532

Offset: 1

R. K. Guy, Dec 02 2004

No other terms below 50000. - Serge Batalov, Jul 23 2017

R. K. Guy, Unsolved Problems In Number Theory, B44.

Yves Gallot, Is the number of primes (of half the left handed factorials) finite?.
Eric Weisstein's World of Mathematics, Left Factorial
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A014288, Left factorials: A003422.

See A124375 for another version.

s = 1; Do[s = s + n!; If[ PrimeQ[s/2], Print[n + 1]], {n, 10^3}] (* Robert G. Wilson v, Dec 02 2004 *)

When A014288(n-1) is prime.

a(14) from Robert G. Wilson v, Dec 02 2004
a(15)=1796 from Ray Chandler, Dec 02 2004
a(17) from T. D. Noe, Dec 04 2004
Corrected by adding a(16)=3733 from Eric W. Weisstein, Oct 29 2005
a(18)=9316 from Eric W. Weisstein, Dec 27 2005
a(19)=12221 from Eric W. Weisstein, Oct 19 2006
a(20)=41532 from Serge Batalov, Jul 22 2017

A124374 Primes of the form !(k + 1)/2 = Sum_{i=0..k} i!/2.

Original entry on oeis.org

2, 5, 17, 2957, 23117, 204557, 2018957, 4578979328975537786697650470157, 12572230784049013026617689884981971446439568309146114097251787122217783800812199225999909965168264460210470157

Offset: 1

Alexander Adamchuk, Oct 28 2006

nonn

Sum_{i=0..k} i! = k! + !k = A003422(k+1), where !k is left factorial !k = Sum_{i=0..k-1} i! = A003422(k). Left factorials are even for k > 1. Corresponding numbers k such that Sum_{i=0..k} i!/2 = A003422(k+1)/2 is prime are listed in A124375(n) = {2, 3, 4, 7, 8, 9, 10, 29, 75, 162, 270, 272, 353, ...}.

Hisanori Mishima, Factorizations of many number sequences.
Eric Weisstein's World of Mathematics, Left Factorial.

Cf. A003422, A100614, A124375.

f=0;Do[f=f+n!;If[PrimeQ[f/2],Print[{n,f/2}]],{n,0,353}]

a(n) = A003422(A124375(k) + 1)/2.

cp's OEIS Frontend

A100614 Numbers n such that (!n)/2 is prime, where !n = Sum_{k=0..n-1} k!.

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