A000946 -id:A000946 - cp's OEIS frontend

A057208 Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.

5, 29, 1237, 32171803229, 829, 405565189, 14717, 39405395843265000967254638989319923697097319108505264560061, 282860648026692294583447078797184988636062145943222437, 53, 421, 13, 109, 4133, 6476791289161646286812333, 461, 34549, 453690033695798389561735541

Offset: 1

Labos Elemer, Oct 09 2000

nonn

			a(3) = 1237 = 8*154 + 5 is the smallest suitable prime divisor of (5*29)*5*29 + 4 = 21029 = 17*1237. (Although 17 is the smallest prime divisor, 17 is not congruent to 5 modulo 8.)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.

P. G. L. Dirichlet, Supplement VI: Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, 1871, 24 pages.

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A007521, A057204-A057208.

a={5}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4+q^2][[All,1]],Mod[#,8]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

lista(nn) = {v = vector(nn); v[1] = 5; print1(v[1], ", "); for (n=2, nn, f = factor(4 + prod(k=1, n-1, v[k])^2); for (k=1, #f~, if (f[k, 1] % 8 == 5, v[n] = f[k,1]; break);); print1(v[n], ", "););} \\ Michel Marcus, Oct 27 2014

More terms from Sean A. Irvine, Oct 26 2014

A051334 Euclid-Mullin sequence (A000945) with initial value a(1)=8191 instead of a(1)=2.

Original entry on oeis.org

8191, 2, 3, 7, 53, 1399, 5, 19, 646843, 26945441, 109, 443, 90670999, 280460690293140589, 907, 16293787, 3655513, 499483, 131, 21067, 143797, 54540542259000816707816058313971443, 392963, 977, 11, 5021, 179, 439, 353, 34417238589462247, 1193114397863177, 13, 59, 31643, 79399, 73, 43, 16639867

Offset: 1

Labos Elemer

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..59

Cf. A000945, A000946, A005265, A005266.

a[1]=8191; 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]=8191; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v;} /* Anders Hellström, Aug 18 2015 */

More terms from Sean A. Irvine, Sep 20 2012

a(30)-a(38) from Charles R Greathouse IV, Sep 21 2012

A051309 Euclid-Mullin sequence (A000945) with initial value a(1)=11 instead of a(1)=2.

Original entry on oeis.org

11, 2, 23, 3, 7, 10627, 433, 17, 13, 10805892983887, 73, 6397, 19, 489407, 2753, 87491, 18618443, 5, 31, 113, 41, 10723, 35101153, 25243, 374399, 966011, 293821591198219762366057, 234947, 4729, 27953, 3256171, 331, 613, 67, 272646324430637, 34281113, 21050393332691947013, 61, 97

Offset: 1

Labos Elemer

nonn

A. R. Booker and S. A. Irvine, The Euclid-Mullin graph, arXiv preprint arXiv:1508.03039 [math.NT], 2015-2016.

Cf. A000945, A000946, A005265, A005266.

a[1]=11; 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]=11; for(i=2, m, v[i]=lpf(1+prod(j=1, i-1, v[j]))); v;
\\ Anders Hellström, Aug 22 2015

Corrected and extended by Sean A. Irvine, Apr 13 2008

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

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.

A258581 a(1) = 2; for n > 1 if n is even a(n) = gpf(1 + Product_{odd m,m

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 23, 37, 17, 149, 761, 50647, 4799, 411527, 18871308021859, 10312625105789, 17838863896549, 57892815889963361050999657943, 2252973546284243766517, 1849093263449444009859625443689931115519009693

Offset: 1

Anders Hellström, Jul 15 2015

nonn

Anders Hellström, Sage program
Anders Hellström, Mercury program

Cf. A000946, A005266.

gpf(n)=my(v=factor(n)[, 1]); v[#v];
main(size)=my(v=vector(size), i, odd=2, even=1); v[1]=2; for(i=2, size, if(i%2==0, v[i]=gpf(odd+1); even*=v[i], v[i]=gpf(even+1); odd*=v[i])); v;

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

cp's OEIS Frontend

A057208 Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.

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A051334 Euclid-Mullin sequence (A000945) with initial value a(1)=8191 instead of a(1)=2.

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A051309 Euclid-Mullin sequence (A000945) with initial value a(1)=11 instead of a(1)=2.

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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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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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A258581 a(1) = 2; for n > 1 if n is even a(n) = gpf(1 + Product_{odd m,m

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

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