A096177 -id:A096177 - cp's OEIS frontend

A096547 Primes p such that primorial(p)/2 - 2 is prime.

5, 7, 11, 13, 17, 19, 23, 31, 41, 53, 71, 103, 167, 431, 563, 673, 727, 829, 1801, 2699, 4481, 6121, 7283, 9413, 10321, 12491, 17807, 30307, 31891, 71917, 172517

Offset: 1

Hugo Pfoertner, Jun 27 2004

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

a(32) > 180000. - Tyler Busby, Mar 29 2024

			Prime 7 is a term because primorial(7)/2 - 2 = A034386(7)/2 - 2 = 2*3*5*7/2 - 2 = 103 is prime.

Cf. A070826, A096177 primes p such that primorial(p)/2+2 is prime, A096178 primes of the form primorial(p)/2+2, A014545 primorial primes, A087398.

Cf. A034386.

b:= proc(n) b(n):= `if`(n=0, 1, `if`(isprime(n), n, 1)*b(n-1)) end:
q:= p-> isprime(p) and isprime(b(p)/2-2):
select(q, [$1..500])[];

k = 1; Do[k *= Prime[n]; If[PrimeQ[k - 2], Print[Prime[n]]], {n, 2, 3276}] (* Ryan Propper, Oct 25 2005 *)
Prime[#]&/@Flatten[Position[FoldList[Times,Prime[Range[1000]]]/2-2,?PrimeQ]] (* _Harvey P. Dale, Jun 09 2023 *)

5 more terms from Ryan Propper, Oct 25 2005

a(29)-a(31) from Tyler Busby, Mar 16 2024

A087398 Primes of the form primorial(P(k))/2-2.

Original entry on oeis.org

13, 103, 1153, 15013, 255253, 4849843, 111546433, 100280245063, 152125131763603, 16294579238595022363, 278970415063349480483707693, 11992411764462614086353260819346129198103, 481473710367991963528473107950567214598209565303106537707981745633

Offset: 1

Cino Hilliard, Oct 21 2003

nonn

Twinmorial numbers are the partial products of twin primes. Sum of reciprocals = 0.08756985926348207565388288916..

The next term (a(14)) has 174 digits. - Harvey P. Dale, Mar 30 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..19

Cf. A096177 primes k such that primorial(k)/2+2 is prime, A096178 primes of the form primorial(k)/2+2, A096547 Primes k such that primorial(k)/2-2 is prime, A067024 smallest p+2 that has n distinct prime factors, A014545 primorial primes.

Select[#/2-2&/@Rest[FoldList[Times,1,Prime[Range[100]]]],PrimeQ] (* Harvey P. Dale, Mar 30 2013 *)

twimorial(n) = { s=0; p=3; forprime(x=5,n, if(isprime(x-2),c1++); p=p*x; if(isprime(p-2), print1(p-2","); c2++; s+=1.0/(p-2); ) ); print(); print(s) }

v=[];pr=1; forprime(p=3,2357,pr*=p; if(ispseudoprime(pr-2),v=concat(v,pr-2))) \\ Charles R Greathouse IV, Feb 14 2011

Twins 3*5 = 15 = p+2. p=13.

Description corrected by Hugo Pfoertner, Jun 25 2004

One more term (a(13)) added by Harvey P. Dale, Mar 30 2013

A096178 Primes of the form primorial(p)/2+2.

Original entry on oeis.org

3, 5, 17, 107, 15017, 3234846617, 100280245067, 3710369067407, 307444891294245707, 961380175077106319537, 139867498408927468089138080936033904837498617

Offset: 1

Hugo Pfoertner, Jun 27 2004

nonn

Primes of the form A070826(n)+2.

			a(4) = 107 because 107 is a prime of the form primorial(7)/2 + 2 = A070826(4) + 2 = 2*3*5*7/2 + 2.

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A070826, A096177 (primorial(p)/2+2 is prime), A096547 (primorial(p)/2-2 is prime), A067024 (smallest p+2 that has n distinct prime factors), A014545 (primorial primes), A087398.

for(n=1,30,p=prod(k=1,n,prime(k))/2+2;if(ispseudoprime(p),print1(p,", "))) \\ Hugo Pfoertner, Dec 26 2019

a(n) = A070826(A096177(n)) + 2. - Amiram Eldar, Dec 26 2019

a(1) inserted by Amiram Eldar, Dec 26 2019

A178641 Primes p such that primorial(p)/2 + 2 is composite.

Original entry on oeis.org

11, 17, 19, 23, 41, 43, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 31 2010

nonn

			3*5*7*11 + 2 = 13*89 is composite.

Cf. A002110, A096177, A178642, A178648.

pp=1;lst={};Do[p=Prime[n];pp*=p;If[ !PrimeQ[pp+2],AppendTo[lst,p]],{n,2,2*5!}];lst

A178642 Primes p such that primorial(p)/2 - 2 is not prime.

Original entry on oeis.org

3, 29, 37, 43, 47, 59, 61, 67, 73, 79, 83, 89, 97, 101, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 31 2010

nonn

			3*5*7*11*13*17*19*23*29 - 2 = 43*167*450473 is composite.

Cf. A002110, A096177, A096547, A178641.

pp=1;lst={};Do[p=Prime[n];pp*=p;If[ !PrimeQ[pp-2],AppendTo[lst,p]],{n,2,2*5!}];lst
Transpose[Select[With[{pros=Rest[FoldList[Times,1,Prime[Range[100]]]]}, Table[ {Prime[n], pros[[n]]},{n,100}]],!PrimeQ[Last[#]/2-2]&]][[1]] (* Harvey P. Dale, Mar 02 2012 *)

A082603 a(n) is the first prime greater than a(n-1) such that a(n)*a(n-1)+2 is a prime, with a(1)=3.

Original entry on oeis.org

3, 5, 7, 11, 19, 23, 37, 47, 73, 227, 241, 251, 271, 317, 367, 563, 607, 641, 727, 761, 829, 1091, 1117, 1223, 1249, 1451, 1579, 1601, 1627, 1721, 1741, 1787, 1873, 1877, 1933, 1973, 2017, 2087, 2137, 2153, 2287, 2351, 2521, 2687, 2707, 2741, 2851, 3041, 3121, 3137, 3181, 3191, 3361, 3371, 3457, 3461, 3541, 3557, 3607, 3701, 3877, 3881, 3907, 3947

Offset: 1

Jon Perry, May 23 2003

			a(3)=7, as 3 + 2 = 5, 3*5 + 2 = 17, 5*7 + 2 = 37, etc.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A039726, A082603, A096177.

f[s_List] := Block[{p = pp = s[[-1]]}, While[p = NextPrime@p; !PrimeQ[ p*pp + 2],]; Append[s, p]]; Nest[f, {3}, 63] (* Robert G. Wilson v, Nov 08 2010 *)
fpg[p1_]:=Module[{p2=NextPrime[p1]},While[!PrimeQ[p1 p2+2],p2=NextPrime[ p2]];p2]; NestList[fpg,3,70] (* Harvey P. Dale, May 31 2021 *)

{ vp=vector(20); vp[1]=3; vc=1; vpt=3; print1(3","); for (vc=2,20, forprime (p=vp[vc-1]+1,10000, if (isprime(p*vp[vc-1]+2),vp[vc]=p;vpt*=p; print1(vp[vc]","); break))) }

{v=3;print1(3",");forprime(p=5,10000,vp=v*p;if(isprime(vp+2),v=vp;print1(p",")))} \\ Zak Seidov, Nov 07 2010

Definition and example corrected to match the sequence and PARI programming, two cross references added, and sequence extended by Robert G. Wilson v, Nov 08 2010

A178648 Primes p such that primorial(p)/2 +- 2 are primes.

Original entry on oeis.org

5, 7, 13, 31

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 31 2010

No further terms up to the 500th prime, i.e., 3571. - Harvey P. Dale, May 09 2023

			3*5 = 15; 15-2 and 15+2 are primes.

Carlos Rivera, Puzzle 974. More twins like these..., The Prime Puzzles & Problems Connection.

Cf. A070826, A178641, A178642.

Intersection of A096177 and A096547.

pp=1;lst={};Do[p=Prime[n];pp*=p;If[PrimeQ[pp-2]&&PrimeQ[pp+2],Print[Date[],p];AppendTo[lst,p]],{n,2,4!}];lst
Module[{nn=15,pr,pm},pr=Prime[Range[nn]];pm=FoldList[Times,pr];Select[Thread[ {pr,pm}],AllTrue[ #[[2]]/2+{2,-2},PrimeQ]&]][[;;,1]] (* Harvey P. Dale, May 09 2023 *)

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A096547 Primes p such that primorial(p)/2 - 2 is prime.

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A087398 Primes of the form primorial(P(k))/2-2.

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A096178 Primes of the form primorial(p)/2+2.

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A178641 Primes p such that primorial(p)/2 + 2 is composite.

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A178642 Primes p such that primorial(p)/2 - 2 is not prime.

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A082603 a(n) is the first prime greater than a(n-1) such that a(n)*a(n-1)+2 is a prime, with a(1)=3.

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A178648 Primes p such that primorial(p)/2 +- 2 are primes.

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