cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Labos Elemer, Oct 09 2000

Keywords

Examples

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

References

  • 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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

More terms from Phil Carmody, Sep 18 2005
Terms corrected and extended by Sean A. Irvine, Oct 23 2014

A102927 Partial products of A102926.

Original entry on oeis.org

2, 6, 30, 870, 9570, 66990, 870870, 32222190, 547777230, 43274401170, 995311226910, 4109640055911390, 793160530790898270, 2056665256340799214110, 207723190890420720625110
Offset: 1

Views

Author

Marc LeBrun, Jan 19 2005

Keywords

Comments

Terms up to a(111) are known from the b-file for A102926. Term a(112) requires us to find the smallest prime factor of at least one of the C472's a(111)+/-1. The companion sequence is listed on the OEIS wiki page "OEIS sequences needing factors" and the corresponding thread on mersenneforum.org. - M. F. Hasler, Mar 21 2013

Examples

			a(4) = 2*3*5*29 = 870.

Crossrefs

Cf. A000945, A005265, A102926.

Formula

Product A102926(k), 1<=k<=n.

A260218 a(1) = 2; for n > 1 if n is even a(n) = spf(1 + Product_{odd m,m

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 257, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 65537, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 97, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 641, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3
Offset: 1

Views

Author

Anders Hellström, Jul 19 2015

Keywords

Links

Crossrefs

Cf. A000945, A005265, A007978, A053669, A075019, A258581.

Programs

  • Mathematica
    f[n_] := Block[{a = {2}, k, m}, Do[If[EvenQ@ k, AppendTo[a, FactorInteger[Product[a[[m]], {m, 1, k - 1, 2}] + 1][[1, 1]]], AppendTo[a, FactorInteger[Product[a[[m]], {m, 2, k - 1, 2}] + 1][[1, 1]]]], {k, 2, n}]; a]; f@ 80 (* Michael De Vlieger, Jul 20 2015 *)
  • PARI
    spf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m), i, odd=2, even=1); v[1]=2; for(i=2, m, if(i%2==0, v[i]=spf(odd+1); even*=v[i], v[i]=spf(even+1); odd*=v[i])); v; /* Anders Hellström, Jul 19 2015 */
  • PARI
    A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
memoA260218 = Map();
A260218(n) = if(1==n,2,if(mapisdefined(memoA260218,n),mapget(memoA260218,n), my(k, m, v = if(!(n%2), k=1; m=1; while(kA260218(k); k += 2); A020639(m+1), k=2; m=1; while(kA260218(k); k += 2); A020639(m+1))); mapput(memoA260218,n,v); (v))); \\ (An incrementally memoized version). Antti Karttunen, Sep 30 2018

Formula

It appears that for odd k, a(k) = 2 and for even k, a(k) = A002586(k/2). - Michel Marcus, Jul 20 2015

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

Views

Author

Anders Hellström, Aug 28 2015

Keywords

Comments

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

Links

Crossrefs

Cf. A000945, A000946, A005265, A005266.

Programs

  • Mathematica
    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 *)
  • PARI
    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;

Extensions

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

Views

Author

Labos Elemer

Keywords

Crossrefs

Cf. A000945, A000946, A005265, A005266.

Programs

  • Mathematica
    a[1]=89; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
  • PARI
    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

Extensions

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

A265009 a(1)=3; for n>1, if n is odd a(n) = spf(Product_{k=1..n-1}(a(k))+1) else a(n) = spf(Product_{k=1..n-1}(a(k))-1), where spf is "smallest prime factor".

Original entry on oeis.org

3, 2, 7, 41, 1723, 5, 14835031, 220078129935929, 241, 23, 79, 101, 23291, 11, 223, 122386298896281959929015788890561251765109069, 38803, 17, 8209, 59, 199, 3340389589, 11527, 13, 47939, 1163, 599, 27198087874669514440553, 181936481, 31, 383, 9623, 739, 33287, 1061, 6493520653, 587, 709, 6548057, 1823, 361789, 20183
Offset: 1

Views

Author

Anders Hellström, Nov 30 2015

Keywords

Crossrefs

Cf. A000945, A005265.

Programs

  • Mathematica
    a[1] = 3; a[n_] := a[n] = FactorInteger[ Product[a[k], {k, n - 1}] + If[OddQ@ n, 1, -1]][[1, 1]]; Array[a, {16}] (* Michael De Vlieger, Nov 30 2015 *)
  • PARI
    spf(n)=my(f=factor(n)[1, 1]); f
first(m)=my(v=vector(m)); v[1]=3; for(i=2, m,;v[i]=spf((-1)^(i+1)+prod(j=1, i-1, v[j]))); v

Extensions

a(20)-a(42) from Hans Havermann, Dec 06 2015
Previous Showing 41-46 of 46 results.

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