A212281 -id:A212281 - cp's OEIS frontend

A212282 Minimal m >= 1 such that n!/m-1 is prime.

1, 1, 2, 1, 1, 4, 2, 4, 5, 1, 10, 1, 4, 3, 6, 4, 8, 6, 9, 25, 4, 28, 12, 11, 9, 10, 3, 1, 2, 1, 1, 3, 34, 5, 9, 1, 35, 24, 2, 5, 3, 24, 16, 7, 5, 32, 23, 6, 13, 13, 16, 7, 6, 20, 7, 31, 6, 94, 22, 26, 78, 9, 21, 8, 15, 11, 88, 11, 7, 58, 64, 56, 16

Offset: 3

Vladimir Shevelev, Feb 14 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 3..1002

Cf. A212281.

a[n_] := Block[{f = n!, j=1}, While[! PrimeQ[Floor[f/j]-1], j++]; j]; v = a /@ Range[3,75] (* Giovanni Resta, Feb 14 2013 *)

a(12)-a(75) from Giovanni Resta, Feb 14 2013

A212321 Minimal m >= 1 such that floor((2*n-1)!!/m) + 2 is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 9, 5, 5, 3, 1, 1, 7, 1, 13, 11, 15, 21, 23, 15, 17, 5, 25, 25, 7, 85, 9, 25, 1, 21, 9, 13, 17, 5, 49, 39, 45, 1, 43, 51, 145, 17, 143, 85, 19, 81, 63, 43, 47, 77, 23, 1, 3, 21, 17, 47, 37, 3, 7, 67, 23, 23, 25, 45, 81, 5, 39, 145, 71, 21, 41

Offset: 1

Vladimir Shevelev, Feb 14 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A212281, A212282.

a:= proc(n) local m;
      for m while not isprime(iquo(doublefactorial(2*n-1), m)+2)
      do od; m
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 18 2013

a[n_] := Module[{m}, For[m = 1, True, m++, If[PrimeQ[Floor[(2n-1)!!/m]+2], Return[m]]]];
Array[a, 100] (* Jean-François Alcover, Oct 28 2020 *)

More terms from Alois P. Heinz, Feb 18 2013

A212636 Minimal m >= 1 such that floor((2*n - 1)!!/m) - 2 is prime.

Original entry on oeis.org

1, 1, 3, 3, 8, 1, 1, 1, 5, 7, 19, 7, 5, 15, 7, 17, 5, 3, 9, 11, 63, 9, 5, 1, 53, 27, 51, 11, 3, 11, 13, 15, 17, 35, 1, 17, 21, 13, 139, 61, 3, 13, 1, 7, 147, 23, 123, 47, 41, 35, 11, 39, 69, 21, 123, 29, 27, 49, 3, 9, 37, 171, 57, 1, 31, 37, 5, 61, 27, 31, 53

Offset: 3

Vladimir Shevelev, Feb 14 2013

nonn

			(2*6-1)!! = 11!! = 11 * 9 * 7 * 5 * 3 * 1 = 10395. Floor(10395/1) - 2 = 10395 - 2 = 10393 =  19 * 547 is not prime, and floor(10395/2) - 2 = 5197 - 2 = 5195 = 5 * 1039 is not prime, but floor(10395/3) - 2 = 3465 - 2 = 3463 is prime, so a(6) = 3.

Cf. A212281, A212282, A212321.

a:= proc(n) local m;
      for m while not isprime(iquo(doublefactorial(2*n-1), m)-2)
      do od; m
    end:
seq(a(n), n=3..70);  #  Alois P. Heinz, Feb 18 2013

a[n_] := Module[{m}, For[m = 1, True, m++, If[PrimeQ[Floor[(2n-1)!!/m]-2], Return[m]]]];
Table[a[n], {n, 3, 73}] (* Jean-François Alcover, May 24 2025 *)

More terms from Alois P. Heinz, Feb 18 2013

A212637 Minimal m>=0 such that (2*n-1)!! + 2^m is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 4, 3, 2, 6, 1, 1, 3, 1, 2, 7, 5, 11, 16, 2, 3, 2, 4, 4, 11, 6, 4, 33, 1, 12, 18, 3, 20, 5, 38, 17, 2, 1, 18, 3, 14, 26, 11, 14, 63, 3, 2, 13, 110, 44, 34, 1, 22, 44, 114, 37, 43, 39, 5, 13, 12, 57, 41, 12, 18, 13, 9, 8, 13, 52, 26, 78, 37, 14

Offset: 1

Vladimir Shevelev, Feb 14 2013

nonn

Cf. A212281, A212282, A212321, A212636.

a:= proc(n) local m;
      for m from 0 while not isprime(doublefactorial(2*n-1)+2^m)
      do od; m
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 18 2013

a[n_] := Block[{f = (2*n - 1)!!, m = 0}, While[! PrimeQ[f + 2^m], m++]; m]; a/@Range[70] (* Giovanni Resta, Feb 15 2013 *)

a(9)-a(70) from Giovanni Resta, Feb 15 2013

A212648 Minimal m >= 0 such that (2*n-1)!! - 2^m is prime.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 1, 1, 6, 9, 2, 6, 6, 8, 16, 2, 15, 6, 3, 3, 5, 7, 26, 1, 8, 16, 18, 7, 14, 12, 9, 5, 14, 8, 1, 32, 7, 2, 2, 3, 53, 8, 1, 3, 10, 10, 20, 8, 25, 20, 2, 23, 7, 13, 21, 87, 16, 76, 35, 30, 18, 12, 7, 1, 117, 36, 40, 57, 25, 3, 5, 47, 62

Offset: 2

Vladimir Shevelev, Feb 14 2013

nonn

Robert Israel, Table of n, a(n) for n = 2..1000

Cf. A212281, A212282, A212321, A212636, A212637.

f:= proc(n) local m,t;
  t:= doublefactorial(2*n-1);
  for m from 0 do
    if isprime(t - 2^m) then return m fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Jul 20 2023

a[n_] := Block[{f=(2*n-1)!!, m=0}, While[! PrimeQ[f - 2^m], m++]; m]; a /@ Range[2, 75] (* Giovanni Resta, Feb 14 2013 *)

a(8)-a(75) from Giovanni Resta, Feb 14 2013

A213366 Minimal of max(m,k) (m>=1,k>=0) such that (2*n-1)!!/m-2^k is prime.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 1, 1, 5, 3, 2, 6, 5, 3, 5, 2, 3, 3, 3, 3, 5, 5, 3, 1, 7, 5, 8, 7, 3, 5, 5, 5, 5, 7, 1, 9, 7, 2, 2, 3, 3, 7, 1, 3, 10, 7, 5, 4, 11, 9, 2, 9, 7, 9, 9, 13, 13, 9, 3, 9, 5, 12, 7, 1, 5, 7, 5, 9, 6, 3, 5, 5, 11, 10, 1, 5, 5, 7, 3, 3, 3, 8, 9, 2, 6, 7, 4, 7, 9, 5, 4, 6, 7, 5, 10, 15, 4, 9, 11, 15, 9, 21, 13, 7, 11, 7, 8, 7, 8, 12, 7, 4, 12, 5, 7, 20, 9, 5, 21, 2, 1, 3, 9, 5, 13, 16, 13, 9, 4, 15, 11, 13, 5, 2, 9, 7, 11, 11, 11, 17, 9, 8, 9, 15, 15, 2, 12, 3

Offset: 2

Vladimir Shevelev, Feb 15 2013

nonn

Cf. A212281, A212282, A212321, A212636, A212637, A212648.

a(n)=my(m=-1, P=prod(i=2, n, 2*i-1)); while(m+=2, for(k=m-2, m-1, forstep(M=1, m-2, 2, if(ispseudoprime(P/M-2^k), return(k)))); for(k=0, m, if(ispseudoprime(P/m-2^k), return(m)))) \\ Charles R Greathouse IV, Feb 27 2013

Terms beginning with a(10) from Peter J. C. Moses

A240133 Least number k such that n!/k + 1 and n!/k - 1 are twin primes.

Original entry on oeis.org

1, 2, 2, 3, 12, 12, 48, 8, 5, 54, 44, 6, 24, 39, 6, 81, 20, 30, 19, 25, 380, 28, 264, 50, 52, 250, 35, 385, 20, 77, 182, 405, 77, 605, 143, 720, 144, 722, 96, 713, 46, 403, 98, 4508, 90, 77, 560, 806, 64, 665, 2376, 1785, 893, 1235, 4278, 159, 66, 1326, 1806, 429, 475

Offset: 3

Derek Orr, Apr 02 2014

nonn

			6!/1-1 (719) and 6!/1+1 (721) are not both prime. 6!/2-1 (359) and 6!/2+1 (361) are not both prime. 6!/3-1 (239) and 6!/3+1 (241) are both prime. thus, a(6) = 3.

Giovanni Resta, Table of n, a(n) for n = 3..350

Cf. A212281, A212282.

lnk[n_]:=Module[{nf=n!,k=1},While[!AllTrue[nf/k+{1,-1},PrimeQ],k++];k]; Array[lnk,70,3] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 18 2015 *)

f(n)=for(k=1,n!,if(floor(n!/k-1)==n!/k-1 && floor(n!/k+1)==n!/k+1,if(ispseudoprime(n!/k-1) && ispseudoprime(n!/k+1),return(k))))

A213512 Minimal of max(m,k) (m>=1,k>=0) such that (2*n-1)!!/m+2^k is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 2, 3, 1, 1, 3, 1, 2, 4, 5, 5, 5, 2, 3, 2, 4, 4, 7, 5, 4, 7, 1, 9, 6, 3, 6, 5, 13, 10, 2, 1, 7, 3, 6, 5, 5, 5, 9, 3, 2, 5, 9, 7, 11, 1, 3, 5, 10, 5, 7, 3, 3, 5, 9, 4, 5, 11, 8, 5, 9, 8, 11, 9, 9, 13, 9, 11, 11, 9, 5, 3, 11, 11, 3, 7, 3, 9, 1, 10, 7, 9, 3, 7, 7, 11, 9, 6, 9, 16, 5, 6, 9, 5, 9, 15, 11, 12, 3, 19, 7, 7, 3, 1, 19, 7, 9, 11, 4, 19, 5, 9, 7, 11, 13, 11, 1, 7, 15, 7, 15, 18, 11, 7, 5, 19, 15, 12, 7, 19, 14, 9, 7, 11, 3, 13, 10, 13, 16, 11, 9, 11, 7, 3

Offset: 1

Vladimir Shevelev, Feb 15 2013

nonn

Cf. A212281, A212282, A212321, A212636, A212637, A212648, A213366.

a(n)=my(m=-1,P=prod(i=2,n,2*i-1)); while(m+=2, for(k=m-2,m-1, forstep(M=1,m-2,2, if(ispseudoprime(P/M+2^k),return(k)))); for(k=0,m, if(ispseudoprime(P/m+2^k),return(m)))) \\ Charles R Greathouse IV, Feb 27 2013

Terms beginning with a(6) were calculated by Peter J. C. Moses.

cp's OEIS Frontend

A212282 Minimal m >= 1 such that n!/m-1 is prime.

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A212321 Minimal m >= 1 such that floor((2*n-1)!!/m) + 2 is prime.

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A212636 Minimal m >= 1 such that floor((2*n - 1)!!/m) - 2 is prime.

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A212637 Minimal m>=0 such that (2*n-1)!! + 2^m is prime.

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A212648 Minimal m >= 0 such that (2*n-1)!! - 2^m is prime.

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A213366 Minimal of max(m,k) (m>=1,k>=0) such that (2*n-1)!!/m-2^k is prime.

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PARI

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A240133 Least number k such that n!/k + 1 and n!/k - 1 are twin primes.

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Programs

Mathematica

PARI

A213512 Minimal of max(m,k) (m>=1,k>=0) such that (2*n-1)!!/m+2^k is prime.

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PARI

Extensions