A051335 -id:A051335 - cp's OEIS frontend

A094461 a[n] is the 5th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

13, 13, 331, 13, 7, 6163, 7, 571, 13, 10267, 23, 31, 7, 13, 17, 7, 3, 7, 5227, 43, 7, 2371, 7, 61, 19, 3, 7, 13, 3271, 13, 5, 37, 4111, 43, 3, 13, 47, 7, 5011, 360187, 7, 73, 13, 22003, 23, 7, 8863, 5, 7, 6871, 181, 193, 7, 7, 11, 139, 3, 7, 1297, 73, 7, 7, 31, 3, 7

Offset: 1

Labos Elemer, May 06 2004

nonn

			First term is p[n], 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is here in A094461;
6th, 7th terms in A094462, A094463;

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=5};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094463 a(n) is the 7th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

5, 5, 199, 5, 433, 1601, 31, 457, 7109609443, 5, 7, 127, 71, 5, 7, 2620003, 4583, 1217, 5, 67, 6729871, 39334891, 5, 53, 461, 449885311, 1511, 197, 7, 22008559, 19, 1249, 7, 7, 3217, 7, 7, 3931, 7, 110663370509047, 375155719, 29, 28529671, 23, 24603331

Offset: 1

Labos Elemer, May 06 2004

nonn

			First term is p(n), 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is in A094461;
6th-7th terms in A094462, A094463;

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=6};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094462 a(n) is the 6th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

53, 53, 19, 53, 10627, 7, 3571, 271, 84319, 7, 47059, 7, 47, 53, 23971, 11, 13, 5, 7, 201499, 5, 7, 67, 13, 7, 21211, 5, 29, 10696171, 11, 149, 971, 16896211, 11, 58111, 17, 11, 75307, 25105111, 853, 139, 7, 5, 613, 181, 23, 13, 29, 13, 19, 53, 47, 5, 11, 84811

Offset: 1

Labos Elemer, May 06 2004

nonn

			First term is p(n), 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is in A094461;
6th-7th terms in A094462, A094463;

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=6};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A057206 Primes of the form 6k+5 generated recursively: a(1)=5; a(n) = min{p, prime; p mod 6 = 5; p | 6Q-1}, where Q is the product of all previous terms in the sequence.

Original entry on oeis.org

5, 29, 11, 1367, 13082189, 89, 59, 29819952677, 91736008068017, 17, 887050405736870123700827, 688273423680369013308306870159348033807942418302818522537, 74367405177105011, 12731422703, 1812053

Offset: 1

Labos Elemer, Oct 09 2000

nonn

There are infinitely many primes of the form 6k + 5, and this sequence figures in the classic proof of that fact. - Alonso del Arte, Mar 02 2017

			a(3) = 11 is the smallest prime divisor of the form 6k + 5 of 6 * (5 * 29) - 1 = 6Q - 1 = 11 * 79 = 869.

Dirichlet, P. G. L. (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.

Robert Price, Table of n, a(n) for n = 1..17

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A057204-A057208, A007528.

primes5mod6 = {5}; q = 1;For[n = 2, n <= 10, n++, q = q * Last[ primes5mod6]; AppendTo[primes5mod6, Min[Select[FactorInteger[6 * q - 1][[All, 1]], Mod[#, 6] == 5 &]]];]; primes5mod6 (* Robert Price, Jul 18 2015 *)

main(size)={my(v=vector(size),i,q=1,t);for(i=1,size,t=1;while(!(prime(t)%6==5&&(6*q-1)%prime(t)==0),t++);v[i]=prime(t);q*=v[i]);v;} /* Anders Hellström, Jul 18 2015 */

a(13)-a(17) from Robert Price, Jul 18 2015

A124988 Primes of the form 12k+7 generated recursively. Initial prime is 7. General term is a(n)=Min {p is prime; p divides 3+4Q^2; Mod[p,12]=7}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

7, 199, 7761799, 487, 67, 103, 1482549740515442455520791, 31, 139, 787, 19, 39266047, 1955959, 50650885759, 367, 185767, 62168707

Offset: 1

Nick Hobson, Nov 18 2006

nonn

All prime divisors of 3+4Q^2 are congruent to 1 modulo 6.
At least one prime divisor of 3+4Q^2 is congruent to 3 modulo 4 and hence to 7 modulo 12.
The first six terms are the same as those of A057204.

			a(3) = 1482549740515442455520791 is the smallest prime divisor congruent to 7 mod 12 of 3+4Q^2 = 5281642303363312989311974746340327 = 3562539697 * 1482549740515442455520791, where Q = 7 * 199 * 7761799 * 487 * 67 * 103.

Tyler Busby, Table of n, a(n) for n = 1..21

Cf. A000945, A068229, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={7}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q^2+3][[All,1]],Mod[#,12]==7 &]]];
    ];
a (* Robert Price, Jul 15 2015 *)

A124989 Primes of the form 10k + 9 generated recursively. Initial prime is 19. General term is a(n) = Min_{p is prime; p divides 100Q^2 - 5; p == 9 (mod 10)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

19, 7219, 462739, 509, 129229, 295380580489, 9653956849, 149, 110212292237172705230749846071050188009093377022084806290042881946231583507557298889, 157881589, 60397967745386189, 1429, 79

Offset: 1

Nick Hobson, Nov 18 2006

nonn

100Q^2-5 always has a prime divisor congruent to 9 modulo 10.

			a(3) = 462739 is the smallest prime divisor congruent to 9 mod 10 of 100Q^2-5 = 1881313992095 = 5 * 462739 * 813121, where Q = 19 * 7219.

Robert Price, Table of n, a(n) for n = 1..14

Cf. A000945, A030433, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={19}; q=1;
For[n=2,n<=6,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[100*q^2-5][[All,1]],Mod[#,10]==9&]]];
    ];
a (* Robert Price, Jul 18 2015 *)

A124990 Primes of the form 12k+1 generated recursively. Initial prime is 13. General term is a(n)=Min {p is prime; p divides Q^4-Q^2+1}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

13, 28393, 128758492789, 73, 193, 37, 457, 8363172060732903211423577787181

Offset: 1

Nick Hobson, Nov 18 2006

All prime divisors of Q^4 - Q^2 + 1 are congruent to 1 modulo 12.

			a(3) = 128758492789 is the smallest prime divisor of Q^4 - Q^2 + 1 = 18561733755472408508281 = 128758492789 * 144159296629, where Q = 13 * 28393.

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, NY, Second Edition (1990), p. 63.

Cf. A000945, A068228, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a = {13}; q = 1;
For[n = 2, n ≤ 8, n++,
    q = q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[q^4 - q^2 + 1][[All, 1]],
    Mod[#, 12] == 1 &]]];
    ];
a  (* Robert Price, Jun 25 2015 *)

a(8) from Robert Price, Jun 25 2015

A125039 Primes of the form 8k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^4 + 1}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

17, 1336337, 4261668267710686591310687815697, 41, 4390937134822286389262585915435960722186022220433, 241, 1553, 243537789182873, 97, 27673, 4289, 457, 137201, 73, 337, 569891669978849, 617, 1697, 65089, 1609, 761

Offset: 1

Nick Hobson, Nov 18 2006

nonn

All prime divisors of (2Q)^4 + 1 are congruent to 1 modulo 8.

			a(3) = 4261668267710686591310687815697 is the smallest prime divisor of (2Q)^4 + 1 = 4261668267710686591310687815697, where Q = 17 * 1336337.

G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, NY, (1998), p. 271.

Sean A. Irvine, Table of n, a(n) for n = 1..29

Cf. A000945, A007519, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

More terms from Sean A. Irvine, Apr 09 2015

A217759 Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.

Original entry on oeis.org

3, 7, 43, 19, 6863, 883, 23, 191, 2927, 205677423255820459, 11, 163, 227, 9127, 59, 31, 71, 131627, 2101324929412613521964366263134760336303, 127, 1302443, 4065403, 107, 2591, 21487, 223, 12823, 167, 53720906651, 5452254637117019, 39827899, 11719, 131

Offset: 1

Daran Gill, Mar 23 2013

nonn

Contrast A057207, where all the factors are congruent to 1 (mod 4), here only one is guaranteed to be congruent to 3 (mod 4).

			a(10) is 205677423255820459 because it is the only prime factor congruent to 3 (mod 4) of 4*(3*7*43*19*6863*883*23*191*2927)^2-1 = 5*13*2088217*256085119729*205677423255820459. The four smaller factors are all congruent to 1 (mod 4).

Dirichlet,P.G.L (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 13.

Daran Gill, Table of n, a(n) for n = 1..51
Mersenne Forum, Two new sequences

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002145, A057204-A057208.

A051615 Primes for which A051614 is 3.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 71, 83, 89, 107, 113, 131, 137, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 281, 293, 317, 347, 359, 401, 419, 431, 443, 449, 467, 491, 509, 557, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 719, 743

Offset: 1

Labos Elemer

nonn

Cf. A000945, A051309 - A051335, A005384.

Odd p values such that F(2*p*F(2*p+1)+1)=3, where F(x) is the least prime divisor of x.

cp's OEIS Frontend

A094461 a[n] is the 5th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

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A094463 a(n) is the 7th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

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A094462 a(n) is the 6th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

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A057206 Primes of the form 6k+5 generated recursively: a(1)=5; a(n) = min{p, prime; p mod 6 = 5; p | 6Q-1}, where Q is the product of all previous terms in the sequence.

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A124988 Primes of the form 12k+7 generated recursively. Initial prime is 7. General term is a(n)=Min {p is prime; p divides 3+4Q^2; Mod[p,12]=7}, where Q is the product of previous terms in the sequence.

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A124989 Primes of the form 10*k + 9 generated recursively. Initial prime is 19. General term is a(n) = Min_{p is prime; p divides 100*Q^2 - 5; p == 9 (mod 10)}, where Q is the product of previous terms in the sequence.

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A124990 Primes of the form 12k+1 generated recursively. Initial prime is 13. General term is a(n)=Min {p is prime; p divides Q^4-Q^2+1}, where Q is the product of previous terms in the sequence.

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A125039 Primes of the form 8k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^4 + 1}, where Q is the product of previous terms in the sequence.

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A217759 Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.

A124989 Primes of the form 10k + 9 generated recursively. Initial prime is 19. General term is a(n) = Min_{p is prime; p divides 100Q^2 - 5; p == 9 (mod 10)}, where Q is the product of previous terms in the sequence.