A057204 -id:A057204 - cp's OEIS frontend

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.

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

A125257 Smallest prime divisor of 4n^2+3 that is of the form 6k+1.

Original entry on oeis.org

7, 19, 13, 67, 103, 7, 199, 7, 109, 13, 487, 193, 7, 787, 7, 13, 19, 433, 1447, 7, 19, 7, 13, 769, 2503, 2707, 7, 43, 7, 1201, 3847, 4099, 1453, 7, 4903, 7, 5479, 5779, 2029, 19, 7, 13, 7, 61, 37, 8467, 8839, 7, 13, 7, 3469, 31, 11239, 3889, 7, 12547, 7, 43, 19, 4801

Offset: 1

Nick Hobson, Nov 26 2006

Any prime divisor of 4n^2+3 different from 3 is congruent to 1 modulo 6.
4n^2+3 is never a power of 3 for n > 0; hence a prime divisor congruent to 1 modulo 6 always exists.
a(n) = 7 if and only if n is congruent to 1 or -1 modulo 7.

			The prime divisors of 4*3^2+3=39 are 3 and 13, so a(3) = 13.

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

Nick Hobson, Table of n, a(n) for n = 1..1000

Cf. A057204, A124988.

Table[SelectFirst[FactorInteger[4 n^2+3][[;;,1]],Mod[#,6]==1&],{n,60}] (* Harvey P. Dale, Jan 17 2025 *)

vector(60, n, factor(4*n^2+3)[2-(n^2)%3,1])

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.

A057205 Primes congruent to 3 modulo 4 generated recursively: a(n) = Min_{p, prime; p mod 4 = 3; p|4Q-1}, where Q is the product of all previous terms in the sequence. The initial term is 3.

Original entry on oeis.org

3, 11, 131, 17291, 298995971, 8779, 594359, 59, 151, 983, 19, 38851089348584904271503421339, 2359886893253830912337243172544609142020402559023, 823818731, 2287, 7, 9680188101680097499940803368598534875039120224550520256994576755856639426217960921548886589841784188388581120523, 163, 83, 1471, 34211, 2350509754734287, 23567

Offset: 1

Labos Elemer, Oct 09 2000

nonn

			a(4) = 17291 = 4*4322 + 3 is the smallest prime divisor congruent to 3 (mod 4) of Q = 3*11*131 - 1 = 17291.

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.

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

a={3}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q-1][[All,1]],Mod[#,4]==3&]]];
    ];
a (* Robert Price, Jul 18 2015 *)

More terms from Phil Carmody, Sep 18 2005

Terms corrected and extended by Sean A. Irvine, Oct 23 2014

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 *)

A124991 Primes of the form 10k+1 generated recursively. Initial prime is 11. General term is a(n)=Min {p is prime; p divides (R^5 - 1)/(R - 1); Mod[p,5]=1}, where Q is the product of previous terms in the sequence and R = 5Q.

Original entry on oeis.org

11, 211, 1031, 22741, 41, 15487770335331184216023237599647357572461782407557681, 311, 61, 55172461, 3541, 1381, 2851, 19841, 151, 9033671, 456802301, 1720715817015281, 19001, 71

Offset: 1

Nick Hobson, Nov 18 2006

nonn

All prime divisors of (R^5 - 1)/(R - 1) different from 5 are congruent to 1 modulo 10.

			a(3) = 1031 is the smallest prime divisor congruent to 1 mod 10 of (R^5 - 1)/(R - 1) = 18139194759758381 = 1031 * 17593787351851, where Q = 11 * 211 and R = 5Q.

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

Max Alekseyev, Table of n, a(n) for n=1..34

Cf. A000945, A030430, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={11}; q=1;
For[n=2,n<=6,n++,
    q=q*Last[a]; r=5*q;
    AppendTo[a,Min[Select[FactorInteger[(r^5-1)/(r-1)][[All,1]],Mod[#,10]==1&]]];
    ];
a (* Robert Price, Jul 14 2015 *)

a(20)..a(34) in b-file from Max Alekseyev, Oct 23 2008

A125038 Primes of the form 34k+1 generated recursively. Initial prime is 103. General term is a(n) = Min {p is prime; p divides (R^17 - 1)/(R - 1); p == 1 (mod 17)}, where Q is the product of previous terms in the sequence and R = 17*Q.

Original entry on oeis.org

103, 307, 9929, 187095201191, 76943, 37061, 137, 5615258941637, 302125531, 18089, 613, 409, 9419, 193189

Offset: 1

Nick Hobson, Nov 18 2006

All prime divisors of (R^17 - 1)/(R - 1) different from 17 are congruent to 1 modulo 34.

			a(2) = 307 is the smallest prime divisor congruent to 1 mod 34 of (R^17 - 1)/(R-1) = 7813154903878257490980895975711871949096304270238017 = 307 * 326669135226428664734261 * 77907623430368753779713071, where Q = 103 and R = 17*Q.

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

Cf. A000945, A057204-A057208, A051308-A051335, A124984-A125038, A125037-A125045.

a={103}; q=1;
For[n=2,n<=5,n++,
    q=q*Last[a]; r=17*q;
    AppendTo[a,Min[Select[FactorInteger[(r^17-1)/(r-1)][[All,1]],Mod[#,34]==1 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

a(9)-a(14) from Sean A. Irvine, Jun 27 2011

cp's OEIS Frontend

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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A125257 Smallest prime divisor of 4n^2+3 that is of the form 6k+1.

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PARI

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.

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A057205 Primes congruent to 3 modulo 4 generated recursively: a(n) = Min_{p, prime; p mod 4 = 3; p|4Q-1}, where Q is the product of all previous terms in the sequence. The initial term is 3.

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

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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.

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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.