A005266 -id:A005266 - cp's OEIS frontend

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.

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

A261703 Euclid-Mullin sequence (A000945) with initial value a(1) = 139 instead of a(1) = 2.

Original entry on oeis.org

139, 2, 3, 5, 43, 11, 7, 13, 179489311, 320377, 827, 3895650091, 151, 17, 823, 191, 8648810893, 17548807, 83, 127, 15440491576811767, 106961, 2143, 59, 30689, 71, 26538557132758528345706017618160870665435147, 23, 29, 11721253, 1358804471, 5930216438678837, 39161, 619

Offset: 1

Anders Hellström, Aug 28 2015

nonn

139 was chosen because of the relatively small initial values a(1), .., a(16).

Tyler Busby, Table of n, a(n) for n = 1..65
Planetmath.org, Variants of the Euclid-Mullin sequence
Pollack, Paul; Treviño, Enrique, The Primes that Euclid Forgot.

Cf. A000945, A000946, A005265, A005266.

f[1] = 139; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[1, 1]] ; Table[f[n], {n, 1, 20}] (* Michael De Vlieger, Aug 28 2015, after program at A000945 *)

spf(n)=factor(n)[1,1]
first(m)=my(v=vector(m));v[1]=139;print1(v[1],", ");for(i=2,m,v[i]=spf(1+prod(k=1,i-1,v[k]));;print1(v[i],", "));v;

a(27)-a(33) from Jinyuan Wang, Jul 02 2022

a(34) from Tyler Busby, Oct 12 2023

A051328 Euclid-Mullin sequence (A000945) with initial value a(1)=89 instead of a(1)=2.

Original entry on oeis.org

89, 2, 179, 3, 61, 13, 53, 5, 20086919971, 14911308271, 6016479583547124128827234315771, 9697, 23, 31, 17, 4229, 7937, 22739, 7043, 11, 163, 19, 41, 227399, 73, 4441, 1907, 7, 2647566144823, 47, 337, 211, 1171, 404489, 676721, 71, 653, 4703, 29, 587, 199, 37537, 167, 6893009

Offset: 1

Labos Elemer

Cf. A000945, A000946, A005265, A005266.

a[1]=89; 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]=89; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v \\ Anders Hellström, Dec 06 2015

a(35)-a(44) from Robert Price, Jul 09 2015

cp's OEIS Frontend

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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A261703 Euclid-Mullin sequence (A000945) with initial value a(1) = 139 instead of a(1) = 2.

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A051328 Euclid-Mullin sequence (A000945) with initial value a(1)=89 instead of a(1)=2.

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