A005265 -id:A005265 - cp's OEIS frontend

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.

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

A084598 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is smallest prime factor of (Product_{k = 1..n} a(k)) - 1.

Original entry on oeis.org

2, 3, 5, 29, 11, 7, 13, 37, 32222189, 131, 136013303998782209, 31, 197, 19, 157, 17, 8609, 1831129, 35977, 508326079288931, 487, 10253, 1390043, 18122659735201507243, 25319167, 9512386441, 85577, 1031, 3650460767, 107

Offset: 1

Marc LeBrun, May 31 2003

nonn

Like the Euclid-Mullin sequence A000945, but subtracting rather than adding 1 to the product.

The first 4 terms are identical with A084599. It starts diverging at a(5) because the factorization of 2*3*5*29 - 1 = 869 = 11*79 gives A084598(5)=11 and A084599(5)=79. - Hugo Pfoertner, Mar 31 2004

			a(4) = 29 since 2*3*5 = 30 and 29 is the smallest prime factor of 30-1.

Sean A. Irvine added terms 54 through 61, May 21 2006, giving Table of n, a(n) for n = 1..61
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A000945, A005266, A084598, A084599.

Essentially the same as A005265.

a={2,3}; q=2;
For[n=3,n<=19,n++,
    q=q*Last[a];
    AppendTo[a,Min[FactorInteger[q-1][[All,1]]]];
    ];
a (* Robert Price, Jul 17 2015 *)

More terms from Hugo Pfoertner, May 31 2003, using Dario Alpern's ECM

A051312 Euclid-Mullin sequence (A000945) with initial value a(1)=19 instead of a(1)=2.

Original entry on oeis.org

19, 2, 3, 5, 571, 271, 457, 397, 1123, 23, 103, 42572757267735264511, 313, 17, 16013177, 7951, 1259, 41, 1531, 11, 83, 53, 67, 7, 21397, 13, 1619, 1274209367143, 433, 37, 491, 29, 658837, 135202080527, 163, 587, 31, 2797, 35286479

Offset: 1

Labos Elemer

nonn

Tyler Busby, Table of n, a(n) for n = 1..50 (terms 1..42 from Robert Price)

Cf. A000945, A000946, A005265, A005266.

a[1]=19; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

spf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m)); v[1]=19; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 31 2015

A084599 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is largest prime factor of (Product_{k=1..n} a(k)) - 1.

Original entry on oeis.org

2, 3, 5, 29, 79, 68729, 3739, 6221191, 157170297801581, 70724343608203457341903, 46316297682014731387158877659877, 78592684042614093322289223662773, 181891012640244955605725966274974474087, 547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941

Offset: 1

Marc LeBrun, May 31 2003

nonn

Like the Euclid-Mullin sequence A000946, but subtracting rather than adding 1 to the product.

			a(4)=29 since 2*3*5=30 and 29 is the largest prime factor of 30-1
a(5)=79 since 2*3*5*29=870 and 79 is the largest prime factor of 870-1=869=11*79.

Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A000946, A005265, A084598.

Essentially the same as A005266.

More terms from Hugo Pfoertner, May 31 2003, using Dario Alpern's ECM.

The next term a(15) is not known. It requires the factorization of the 245-digit composite number which remains after eliminating 7 smaller factors.

A051324 Euclid-Mullin sequence (A000945) with initial value a(1)=71 instead of a(1)=2.

Original entry on oeis.org

71, 2, 11, 3, 43, 201499, 67, 5, 487, 19, 967, 13, 131, 17, 3523392679146994953040171, 7, 633046028131441, 197, 1313225762816449, 22441, 29, 7039, 2357, 12264112894355231632110401532068053014661

Offset: 1

Labos Elemer

nonn

Tyler Busby, Table of n, a(n) for n = 1..44 (terms 1..29 from Robert Price)

Cf. A000945, A000946, A005265, A005266.

a[1]=71; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

spf(n)=my(f=factor(n)[1, 1]); f
first(m)=my(v=vector(m)); v[1]=71; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v \\ Anders Hellström, Dec 04 2015

a(24) from Robert Price, Jul 11 2015

A051330 Euclid-Mullin sequence (A000945) with initial value a(1)=97 instead of a(1)=2.

Original entry on oeis.org

97, 2, 3, 11, 19, 7, 461, 719, 5, 1411130344471, 139, 43, 36599, 1097, 17, 104370954301, 23, 13, 59, 41, 83, 196777201807603861, 569, 31, 149, 131, 7408846366410141253195388029, 29, 27017, 192228034594584553, 307, 2677, 73, 263, 389, 10463, 61, 47, 617, 743

Offset: 1

Labos Elemer

Tyler Busby, Table of n, a(n) for n = 1..51 (terms 1..45 from Robert Price)
Andrew R. Booker and Sean A. Irvine, The Euclid-Mullin graph, arXiv preprint arXiv:1508.03039 [math.NT], 2015-2016.

Cf. A000945, A000946, A005265, A005266.

a[1]=97; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

lpf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m)); v[1]=97; for(i=2, m, v[i]=lpf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 31 2015

a(34)-a(45) from Robert Price, Jul 20 2015

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

A102926 Smallest prime factor in product of previous terms +1 or -1.

Original entry on oeis.org

2, 3, 5, 29, 11, 7, 13, 37, 17, 79, 23, 4129, 193, 2593, 101, 19, 39163, 577, 26431, 131, 308798542881428667318174028327605372989, 103, 163, 179, 293, 127, 6287, 683437, 31, 89, 13590243019242466336587034391, 113, 2207, 59, 109, 223, 2351

Offset: 1

Marc LeBrun, Jan 19 2005

nonn

A variant of the Euclid-Mullin construction.

This sequence is listed on the OEIS wiki page "OEIS sequences needing factors" and on the corresponding thread on mersenneforum.org. - M. F. Hasler, Mar 21 2013

			a(5)=11 because 2*3*5*29=870, 869=11*79, 871=13*67.
a(31) = 13590243019242466336587034391 because this is the least prime factor of A102927(30)+1. The least prime factor of A102927(30)-1 is 44989026625856465412069667987. Remarkably, both are 29-digit numbers. - _David Wasserman_, Apr 15 2008

Donovan Johnson, Table of n, a(n) for n = 1..111

Cf. A000945, A000946, A005265, A102927.

spf[{p_,a_}]:=With[{f=FactorInteger[p^2-1][[1,1]]},{p*f,f}]; NestList[ spf,{2,2},36][[All,2]] (* Harvey P. Dale, May 05 2018 *)

a(n) = least prime factor of b(n)^2-1, where b(n) = product a(k), 0A102927.

More terms from Don Reble, Jan 23 2005, corrected Sep 26 2006

Further terms from David Wasserman, Apr 15 2008

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.

A240803 a(n) = 2 + product of first n odd primes.

Original entry on oeis.org

5, 17, 107, 1157, 15017, 255257, 4849847, 111546437, 3234846617, 100280245067, 3710369067407, 152125131763607, 6541380665835017, 307444891294245707, 16294579238595022367, 961380175077106319537, 58644190679703485491637, 3929160775540133527939547, 278970415063349480483707697

Offset: 1

N. J. A. Sloane, Apr 14 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.

Cf. A000945, A000946, A005265, A005266, A240804.

[2+&*[NthPrime(i+1): i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Apr 15 2014

Table[Product[Prime[k + 1], {k, 1, n}] + 2, {n, 1, 30}] (* Vincenzo Librandi, Apr 15 2014 *)

cp's OEIS Frontend

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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A084598 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is smallest prime factor of (Product_{k = 1..n} a(k)) - 1.

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A051312 Euclid-Mullin sequence (A000945) with initial value a(1)=19 instead of a(1)=2.

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A084599 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is largest prime factor of (Product_{k=1..n} a(k)) - 1.

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A051324 Euclid-Mullin sequence (A000945) with initial value a(1)=71 instead of a(1)=2.

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A051330 Euclid-Mullin sequence (A000945) with initial value a(1)=97 instead of a(1)=2.

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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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A102926 Smallest prime factor in product of previous terms +1 or -1.

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