A057208 -id:A057208 - cp's OEIS frontend

A057204 Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.

7, 199, 7761799, 487, 67, 103, 3562539697, 7251847, 13, 127, 5115369871402405003, 31, 697830431171707, 151, 3061, 229, 193, 5393552285540920774057256555028583857599359699, 709, 397, 37, 61, 46168741, 3127279, 181, 122268541

Offset: 1

Labos Elemer, Oct 09 2000

nonn

4*Q^2 + 3 always has a prime divisor congruent to 1 modulo 6.

If we start with the empty product Q=1 then it is not necessary to specify the initial prime. - Jens Kruse Andersen, Jun 30 2014

			a(4)=487 is the smallest prime divisor of 4*Q*Q + 3 = 10812186007, congruent to 1 (mod 6), where Q = 7*199*7761799.

P. G. L. Dirichlet (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.

Sean A. Irvine, Table of n, a(n) for n = 1..48

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002476, A057204-A057208.

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[#,6]==1 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

Q=1;for(n=1,11,f=factor(4*Q^2+3);for(i=1,#f~,p=f[i,1];if(p%6==1,break));print1(p", ");Q*=p) \\ Jens Kruse Andersen, Jun 30 2014

More terms from Nick Hobson, Nov 14 2006

More terms from Sean A. Irvine, Oct 23 2014

A125045 Odd primes generated recursively: a(1) = 3, a(n) = Min {p is prime; p divides Q+2}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 7, 318811, 19, 1747, 12791, 73, 90679, 67, 59, 113, 13, 41, 47, 151, 131, 1301297155768795368671, 20921, 1514878040967313829436066877903, 5514151389810781513, 283, 1063, 3027041, 29, 24040758847310589568111822987, 154351, 89

Offset: 1

Nick Hobson, Nov 18 2006

nonn

The first five terms comprise the known Fermat primes: A019434.

			a(7) = 7 is the smallest prime divisor of 3 * 5 * 17 * 257 * 65537 * 641 + 2 = 2753074036097 = 7 * 11 * 37 * 966329953.

Sean A. Irvine, Table of n, a(n) for n = 1..64

Cf. A000945, A019434, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={3}; q=1;
For[n=2,n<=20,n++,
    q=q*Last[a];
    AppendTo[a,Min[FactorInteger[q+2][[All,1]]]];
    ];
a (* Robert Price, Jul 16 2015 *)

A124984 Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

3, 11, 1091, 1296216011, 2177870960662059587828905091, 76870667, 19, 257680660619, 73677606898727076965233531, 23842300525435506904690028531941969449780447746432390747, 35164737203

Offset: 1

Nick Hobson, Nov 18 2006

nonn

2+Q^2 always has a prime divisor congruent to 3 modulo 8.

			a(3) = 1091 is the smallest prime divisor congruent to 3 mod 8 of 2+Q^2 = 1091, where Q = 3 * 11.

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

Robert Price, Table of n, a(n) for n = 1..15
N. Hobson, Home page (listed in lieu of email address)

Cf. A000945, A007520, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a = {3}; q = 1;
For[n = 2, n ≤ 5, n++,
    q = q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[2 + q^2][[All, 1]], Mod[#,
    8] \[Equal] 3 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

lista(nn) = my(f, q=3); print1(q); for(n=2, nn, f=factor(2+q^2)[, 1]~; for(i=1, #f, if(f[i]%8==3, print1(", ", f[i]); q*=f[i]; break))); \\ Jinyuan Wang, Aug 05 2022

a(10) from Robert Price, Jul 04 2015

a(11) from Robert Price, Jul 05 2015

A125037 Primes of the form 26k+1 generated recursively. Initial prime is 53. General term is a(n) = Min {p is prime; p divides (R^13 - 1)/(R - 1); p == 1 (mod 13)}, where Q is the product of previous terms in the sequence and R = 13*Q.

Original entry on oeis.org

53, 11462027512399586179504472990060461, 25793, 178907, 131, 5669, 3511, 157, 59021, 13070705295701, 547, 79, 424361132339, 126146525792794964042953901, 5889547, 521, 1301, 6249393047, 9829, 2549, 298378081, 29379481, 56993, 1093, 26729

Offset: 1

Nick Hobson, Nov 18 2006

nonn

All prime divisors of (R^13 - 1)/(R - 1) different from 13 are congruent to 1 modulo 26.

			a(2) = 11462027512399586179504472990060461 is the smallest prime divisor congruent to 1 mod 26 of (R^13 - 1)/(R - 1) = 11462027512399586179504472990060461, where Q = 53 and R = 13*Q.

M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.

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

a={53}; q=1;
For[n=2,n<=5,n++,
    q=q*Last[a]; r=13*q;
    AppendTo[a,Min[Select[FactorInteger[(r^13-1)/(r-1)][[All,1]],Mod[#,26]==1 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

More terms from Sean A. Irvine, Jun 24 2011

A124993 Primes of the form 22k+1 generated recursively. Initial prime is 23. General term is a(n) = Min {p is prime; p divides (R^11 - 1)/(R - 1); p == 1 (mod 11)}, where Q is the product of previous terms in the sequence and R = 11*Q.

Original entry on oeis.org

23, 4847239, 2971, 3936923, 9461, 1453, 331, 81373909, 89, 920771904664817214817542307, 353, 401743, 17088192002665532981, 11617

Offset: 1

Nick Hobson, Nov 18 2006

All prime divisors of (R^11 - 1)/(R - 1) different from 11 are congruent to 1 modulo 22.

			a(3) = 2971 is the smallest prime divisor congruent to 1 mod 22 of (R^11-1)/(R-1) =
7693953366218628230903493622259922359469805176129784863956847906415055607909988155588181877
= 2971 * 357405886421 * 914268562437006833738317047149 * 7925221522553970071463867283158786415606996703, where Q = 23 * 4847239, and R = 11*Q.

M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.

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

a={23}; q=1;
For[n=2,n<=2,n++,
    q=q*Last[a]; r=11*q;
    AppendTo[a,Min[Select[FactorInteger[(r^11-1)/(r-1)][[All,1]],Mod[#,11]==1 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

More terms from Max Alekseyev, May 29 2009

A057207 a(1)=5, a(n) is the smallest prime dividing 4*Q^2 + 1 where Q is the product of all previous terms in the sequence.

Original entry on oeis.org

5, 101, 1020101, 53, 29, 2507707213238852620996901, 449, 13, 8693, 1997, 6029, 61, 3181837, 113, 181, 1934689, 6143090225314378441493352126119201470973493456817556328833988172277, 4733, 3617, 41, 68141, 37, 51473, 17, 821, 598201519454797, 157, 9689, 2357, 757, 149, 293, 5261

Offset: 1

Labos Elemer, Oct 09 2000

nonn

Removed redundant mod(p,4) = 1 criterion from definition. By quadratic reciprocity, all factors of 1 + 4Q^2 are congruent to 1 (mod 4). See comments at the end of the b-file for an additional eight terms not proved, but nevertheless highly likely to be correct. - Daran Gill, Mar 23 2013

			a(4)=53 is the smallest prime divisor of 4*(5.101.1020101)^2+1 = 1061522231810040101 = 53*1613*12417062216309.

P. G. L. Dirichlet (1871): Vorlesungen über 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.

Daran Gill, Table of n, a(n) for n = 1..33
Mersenne Forum, Sequence A057207
OEIS wiki, OEIS sequences needing factors

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002144, A057204-A057208.

t = {5}; Do[q = Times @@ t; AppendTo[t, FactorInteger[1 + 4*q^2][[1, 1]]], {6}]; t (* T. D. Noe, Mar 27 2013 *)

Eight more terms, a(9)-a(16), from Max Alekseyev, Apr 27 2009

Seventeen more terms, a(17)-a(33), added by Daran Gill, Mar 23 2013

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

cp's OEIS Frontend

A057204 Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.

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A125045 Odd primes generated recursively: a(1) = 3, a(n) = Min {p is prime; p divides Q+2}, where Q is the product of previous terms in the sequence.

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A124984 Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.

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A125037 Primes of the form 26k+1 generated recursively. Initial prime is 53. General term is a(n) = Min {p is prime; p divides (R^13 - 1)/(R - 1); p == 1 (mod 13)}, where Q is the product of previous terms in the sequence and R = 13*Q.

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A124993 Primes of the form 22k+1 generated recursively. Initial prime is 23. General term is a(n) = Min {p is prime; p divides (R^11 - 1)/(R - 1); p == 1 (mod 11)}, where Q is the product of previous terms in the sequence and R = 11*Q.

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A057207 a(1)=5, a(n) is the smallest prime dividing 4*Q^2 + 1 where Q is the product of all previous terms in the sequence.

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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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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.