A053662 - OEIS

%I #32 Jun 07 2024 08:02:56

%S 3,5,9,21,23,33,39,51,63,65,81,89,95,99,113,131,173,183,191,209,215,

%T 221,239,245,251,261,281,285,299,303,309,315,341,345,363,369,371,393,

%U 411,419,431,443,473,495,509,525,543,545,561,575,593,645,659,683,711

%N Numbers k such that 2k+1 divides k!+1.

%C k+1 divides k!+1 gives primes-1 by Wilson's Theorem. For the present sequence, there are 309 terms below 5000, compared with 669 primes (309/669 = 0.461...). There are 553 terms below 10000, compared with 1229 primes (553/1229 = 0.449...). - _Ed Pegg Jr_, Dec 05 2001

%H G. C. Greubel, <a href="/A053662/b053662.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) >> n log n. - _Charles R Greathouse IV_, Apr 16 2024

%p A053662:=n->`if`(n!+1 mod (2*n+1) = 0, n, NULL): seq(A053662(n), n=1..1000); # _Wesley Ivan Hurt_, Dec 01 2015

%t Drop[Union[Table[If[IntegerQ[(n!+1)/(2n+1)], n], {n, 1, 1000}]], -1] (* _Ed Pegg Jr_, Dec 05 2001 *)

%t Select[Range[1000], Mod[#! +1, 2*# +1] == 0 &] (* _G. C. Greubel_, May 18 2019 *)

%o (PARI) for(n=1,10^3, if((n!+1)%(2*n+1)==0, print1(n,", ")) ) \\ _G. C. Greubel_, May 18 2019

%o (Magma) [n: n in [1..1000] | (Factorial(n)+1) mod (2*n+1) eq 0 ]; // _G. C. Greubel_, May 18 2019

%o (Sage) [n for n in (1..1000) if Mod(factorial(n)+1, 2*n+1)==0 ] # _G. C. Greubel_, May 18 2019

%o (GAP) Filtered([1..1000], n-> (Factorial(n)+1) mod (2*n+1)=0) # _G. C. Greubel_, May 18 2019

%Y Cf. A005097, A006093, A053663.

%K easy,nonn

%O 1,1

%A _Chris K. Caldwell_, Feb 16 2000