A103783 -id:A103783 - cp's OEIS frontend

A073917 Smallest prime which leaves a remainder 1 when divided by primorial(n), i.e., when divided by first n primes.

3, 7, 31, 211, 2311, 120121, 4084081, 106696591, 892371481, 71166625531, 200560490131, 29682952539241, 2129751844690471, 78496567990020181, 8608456956238879741, 97767475431570134191, 9613801750771063195351

Offset: 1

Amarnath Murthy, Aug 18 2002

nonn

Let Pr(n) = the product of first n primes. Then a(n) is the smallest prime of the form k*Pr(n) + 1. k = 1 for first five terms.

Smallest prime p such that the prime factorization of p-1 contains the first n primes. - R. J. Mathar, Jul 03 2012

T. D. Noe, Table of n, a(n) for n = 1..150

Cf. A002110 (primorials), A073915, A103783, A214089.

Cf. A076689 (values of k).

a(n)=if(n<0,0,s=1; while(prime(s)%prod(i=1,n, prime(i))>1,s++); s)

More terms from Vladeta Jovovic, Aug 20 2002

A214089 Least prime p such that the first n primes divide p^2-1.

Original entry on oeis.org

3, 5, 11, 29, 419, 1429, 1429, 315589, 1729001, 57762431, 1724478911, 6188402219, 349152569039, 1430083494841, 390499187164241, 1010518715554349, 18628320726623609, 522124211958421799, 522124211958421799, 5936798290039408015951, 311263131154464891496249

Offset: 1

Robin Garcia, Jul 02 2012

(a(n)^2 - 1) / A002110(n) is congruent to 0 (mod 4). (a(n)^2 - 1) / (4*A002110(n)) = A215085(n). [J. Stauduhar, Aug 03 2012]

a(n) == +1 or -1 (mod prime(i)) for every i=1,2,...,n. The system of congruences x == +1 or -1 (mod prime(i)), i=1,2,...,n, has 2^(n-1) solutions modulo A002110(n) so that a(n) represents the smallest prime in the corresponding residue classes, allowing efficient computation (see PARI program). - Max Alekseyev, Aug 22 2012

			a(5) = 419: 419^2-1 = 175560 = 2^3*3*5*7*11*19 contains the first 5 primes.
a(7) = 1429:  1428=2^2*3*7*17, 1430=2*5*11*13 contains the first 7 primes.
a(8) = 315589: 315589^2-1 = 2^3*3*5*7*11*13*17^2*19*151 contains the first 8 primes.

Max Alekseyev, Table of n, a(n) for n = 1..32

Cf. A073917, A103783.

A214089 := proc(n)
     local m,k,p;
   m:= 2*mul(ithprime(j),j=1..n);
   for k from 1 do
     p:= sqrt(m*k+1);
     if type(p,integer) and isprime(p) then return(p)
     end if
   end do
end proc;
# Robert Israel, Aug 19 2012

f[n_] := Block[{k = 1, p = Times @@ Prime@Range@n}, While[! IntegerQ@Sqrt[4 k*p + 1], k++]; Block[{j = k}, While[! PrimeQ[Sqrt[4 j*p + 1]], j++]; Sqrt[4 j*p + 1]]]; Array[f, 10] (* J. Stauduhar, Aug 18 2012 *)

A214089(n) = {
        local(a,k=4,p) ;
        a=prod(j=1,n,prime(j)) ;
        while(1,
                if( issquare(k*a+1, &p),
                        if(isprime(p),
                                return(p);
                        ) ;
                ) ;
                k+=4;
        ) ;
} ;

{ a(n) = local(B,q); B=prod(i=1,n,prime(i))^2; forvec(v=vector(n-1,i,[0,1]), q=chinese(concat(vector(n-1,i,Mod((-1)^v[i],prime(i+1))),[Mod(1,2)])); forstep(s=lift(q),B-1,q.mod,if(ispseudoprime(s),B=s;break)) ); B } /* Max Alekseyev, Aug 22 2012 */

from itertools import product
from sympy import sieve, prime, isprime
from sympy.ntheory.modular import crt
def A214089(n): return 3 if n == 1 else int(min(filter(isprime,(crt(tuple(sieve.primerange(prime(n)+1)), t)[0] for t in product((1,-1),repeat=n))))) # Chai Wah Wu, May 31 2022

a(15)-a(16) from Donovan Johnson, Jul 25 2012
a(17) from Charles R Greathouse IV, Aug 08 2012
a(18) from Charles R Greathouse IV, Aug 16 2012
a(19) from J. Stauduhar, Aug 18 2012
a(20)-a(32) from Max Alekseyev, Aug 22 2012

A103784 Minimal n that makes primorial P(k)*n-1 prime, k>=2, n>0.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 6, 3, 2, 1, 18, 3, 7, 13, 11, 24, 2, 13, 2, 16, 1, 4, 29, 6, 18, 4, 2, 11, 14, 13, 38, 13, 14, 9, 17, 12, 13, 10, 31, 19, 5, 58, 5, 15, 22, 18, 8, 5, 11, 27, 24, 13, 10, 11, 3, 36, 18, 19, 13, 16, 12, 3, 1, 53, 1, 11, 19, 15, 81, 14, 28, 7, 5, 57, 40, 40, 46, 6, 10

Offset: 2

Lei Zhou, Feb 15 2005

nonn

Minimal n of sequence A103783. Weak conjecture: sequence is defined for all k>=2; strong conjecture: a(k)<=(prime(k))^2;

			P(2)*1-1=5 is prime, so a(2)=1;
P(9)*3-1=669278609 is prime, so a(9)=3;

Cf. A002110, A103515, A103783.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 1; cp = npd*tt - 1; While[(tt <= (Prime[n])^2) && (! (PrimeQ[cp])), tt = tt + 1; cp = npd*tt - 1]; If[tt >= (Prime[n])^2, cp = -cp; tn1 = -tt, tn1 = tt]; Print[tn1]; n = n + 1; npd = npd*Prime[n]]

A177064 Primorial indices j such that P(j)#*2^k - 1 is a lower twin prime for the minimal k selected in A103782.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 30, 96, 148, 171, 201, 246, 274, 294, 467, 543, 603, 614

Offset: 1

Pierre CAMI, Dec 09 2010

nonn

For each j, the sequence A103782 constructs a prime of the form P(j)#*2^k - 1. If this is also a lower twin prime, then j is a term of this sequence.

			P(0)# = 1, P(0)#*2^2 - 1 = 3, P(0)#*2^2 + 1 = 5 twin prime of 5 so a(1)=0;
P(1)# = 1*2, P(1)#*2^1 - 1 = 3, P(1)#*2^1 + 1 = 5 twin prime of 5 so a(2)=1;
P(2)# = 1*2*3, P(2)#*2^1 - 1 = 11, P(2)#*2^1 + 1 = 13 twin prime of 11 so a(3)=2.

Cf. A001359, A103782, A103783, A176994, A177031.

isA001359 := proc(n) isprime(n) and isprime(n+2) ; end proc:
A002110 := proc(n) mul(ithprime(i),i=1..n) ; end proc:
A103782 := proc(n) local m ; for m from 0 do if isprime(A002110(n)*2^m-1) then return m; end if; end do: end proc:
isA177064 := proc(n) A002110(n)*2^A103782(n)-1 ; isA001359(%) ; end proc:
for n from 0 do if isA177064(n) then print(n) ; end if; end do: # R. J. Mathar, Dec 12 2010

{j: A002110(j)*2^A103782(j)-1 in A001359}.

cp's OEIS Frontend

A073917 Smallest prime which leaves a remainder 1 when divided by primorial(n), i.e., when divided by first n primes.

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A214089 Least prime p such that the first n primes divide p^2-1.

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A103784 Minimal n that makes primorial P(k)*n-1 prime, k>=2, n>0.

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Mathematica

A177064 Primorial indices j such that P(j)#*2^k - 1 is a lower twin prime for the minimal k selected in A103782.

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Maple

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