A125045 -id:A125045 - cp's OEIS frontend

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.

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

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

A339344 Lexicographically earliest sequence of odd primes such that the asymptotic density of the numbers which are divisible by at least one of these primes is 1/2.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967311, 1229782942255939601, 88962710886098567818446141338419231, 255302062200114858892457591448999891874349780170241684791167583265041

Offset: 1

Amiram Eldar, Nov 30 2020

nonn

Given a set of prime numbers P, finite or infinite, the set of numbers which are divisible by at least one of the primes in P has an asymptotic density Product_{p in P} (1 - 1/p). If P is finite, then this density is equal to 1/2 only when P = {2}. Otherwise, the density is 1/2 for infinitely many sets P. This sequence is the lexicographically earliest infinite sequence of such primes.
The first 5 terms are the Fermat primes (A019434).
a(10) = 7.455916... * 10^135 is too large to be included in the data section.

Amiram Eldar, Table of n, a(n) for n = 1..12

Cf. A000215, A019434, A080307, A125045, A262228, A339345.

s = {}; r = 1; p = 3; Do[AppendTo[s, p]; r *= 1 - 1/p; p = NextPrime[r/(r - 1/2)], {9}]; s

a(1) = 3, a(n) = nextprime(r(n-1)/(r(n-1) - 1/2)), where r(n) = Product_{k=1..n-1} 1 - 1/a(n).

Product_{n=>1} (1 - 1/a(n)) = 1/2.

A124986 Primes of the form 12k + 5 generated recursively. Initial prime is 5. General term is a(n) = Min_{p is prime; p divides 1 + 4Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

5, 101, 1020101, 53, 29, 2507707213238852620996901, 449, 433361, 401, 925177698346131180901394980203075088053316845914981, 44876921, 17, 173

Offset: 1

Nick Hobson, Nov 18 2006 and Nov 23 2006

nonn

All prime divisors of 1+4Q^2 are congruent to 1 modulo 4.
At least one prime divisor of 1+4Q^2 is congruent to 2 modulo 3 and hence to 5 modulo 12.
The first seven terms are the same as those of A057207.
The next term is known but is too large to include.

			a(8) = 433361 is the smallest prime divisor congruent to 5 mod 12 of 1+4Q^2 = 3179238942812523869898723304484664524974766291591037769022962819805514576256901 = 13 * 433361 * 42408853 * 2272998442375593325550634821 * 5854291291251561948836681114631909089, where Q = 5 * 101 * 1020101 * 53 * 29 * 2507707213238852620996901 * 449.

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

Cf. A000945, A040117, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={5}; q=1;
For[n=2,n<=5,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q^2+1][[All,1]],Mod[#,12]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

A124987 Primes of the form 12k+5 generated recursively. Initial prime is 5. General term is a(n) = Min {p is prime; p divides 4+Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

5, 29, 17, 6076229, 1289, 78067083126343039013, 521, 8606045503613, 15837917, 1873731749, 809, 137, 2237, 17729

Offset: 1

Nick Hobson, Nov 18 2006

Since Q is odd, all prime divisors of 4+Q^2 are congruent to 1 modulo 4.
At least one prime divisor of 4+Q^2 is congruent to 2 modulo 3 and hence to 5 modulo 12.
The first two terms are the same as those of A057208.

			a(3) = 17 is the smallest prime divisor congruent to 5 mod 12 of 4+Q^2 = 21029 = 17 * 1237, where Q = 5 * 29.

Tyler Busby, Table of n, a(n) for n = 1..15

Cf. A000945, A040117, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={5}; q=1;
For[n=2,n<=5,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[q^2+4][[All,1]],Mod[#,12]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

cp's OEIS Frontend

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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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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A339344 Lexicographically earliest sequence of odd primes such that the asymptotic density of the numbers which are divisible by at least one of these primes is 1/2.

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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.

A124986 Primes of the form 12k + 5 generated recursively. Initial prime is 5. General term is a(n) = Min_{p is prime; p divides 1 + 4Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.