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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A094152 a(n) is the position of prime 7 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

3, 3, 15, 1, 5, 6, 5, 24, 10, 6, 7, 6, 5, 4, 7, 5, 3, 5, 6, 16, 5, 6, 5, 28, 6, 3, 5, 36, 7, 15, 4, 15, 7, 7, 8, 7, 7, 5, 7, 14, 5, 6, 19, 16, 17, 5, 4, 12, 5, 8, 10, 17, 5, 5, 8, 10, 3, 5, 7, 30, 5, 5, 20, 3, 5, 6, 6, 4, 9, 9, 3, 9, 5, 6, 8, 8
Offset: 1

Views

Author

Labos Elemer, May 05 2004

Keywords

Examples

			n=8: p(8)=19, the corresponding EM sequence is A051312 in which p=7 arises at the 24th position as follows:
{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}, thus a(8)=24.

Links

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777-A093784.

Extensions

More terms from Sean A. Irvine, Sep 20 2012

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

Views

Author

Labos Elemer, Oct 09 2000

Keywords

Comments

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

Examples

			a(4)=53 is the smallest prime divisor of 4*(5.101.1020101)^2+1 = 1061522231810040101 = 53*1613*12417062216309.

References

  • 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.

Links

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002144, A057204-A057208.

Programs

  • Mathematica
    t = {5}; Do[q = Times @@ t; AppendTo[t, FactorInteger[1 + 4*q^2][[1, 1]]], {6}]; t (* T. D. Noe, Mar 27 2013 *)

Extensions

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

A094153 a(n) is least prime p such that 7 is the n-th term in the Euclid-Mullin sequence starting at p, or 0 if no such prime p exists.

Original entry on oeis.org

7, 0, 2, 43, 11, 13, 31, 149, 347, 23, 439, 223, 461, 173, 5, 71, 197, 1153, 191, 307, 1657, 971, 9473, 19, 2399, 1607, 6781, 89, 9187, 281, 23623, 15077, 25579, 17203
Offset: 1

Views

Author

Labos Elemer, May 05 2004

Keywords

Comments

The sequence is not monotonic. Compare to A093882.
Next term exceeds 50000. - Sean A. Irvine, Jan 12 2012

Examples

			a(5)=11 because p=7 first arises in EM at position 5, which is initiated with 11: {11,2,23,3,7,10627,433}; see A051309.

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777-A093783.

Extensions

Definition clarified, terms corrected and extended by Sean A. Irvine, Apr 15 2011
More terms from Sean A. Irvine, May 22 2011
25579 and 17203 from Sean A. Irvine, Jan 11 2012

A094461 a[n] is the 5th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

13, 13, 331, 13, 7, 6163, 7, 571, 13, 10267, 23, 31, 7, 13, 17, 7, 3, 7, 5227, 43, 7, 2371, 7, 61, 19, 3, 7, 13, 3271, 13, 5, 37, 4111, 43, 3, 13, 47, 7, 5011, 360187, 7, 73, 13, 22003, 23, 7, 8863, 5, 7, 6871, 181, 193, 7, 7, 11, 139, 3, 7, 1297, 73, 7, 7, 31, 3, 7
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p[n], 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is here in A094461;
6th, 7th terms in A094462, A094463;

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=5};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094463 a(n) is the 7th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

5, 5, 199, 5, 433, 1601, 31, 457, 7109609443, 5, 7, 127, 71, 5, 7, 2620003, 4583, 1217, 5, 67, 6729871, 39334891, 5, 53, 461, 449885311, 1511, 197, 7, 22008559, 19, 1249, 7, 7, 3217, 7, 7, 3931, 7, 110663370509047, 375155719, 29, 28529671, 23, 24603331
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p(n), 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is in A094461;
6th-7th terms in A094462, A094463;

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=6};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094462 a(n) is the 6th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

53, 53, 19, 53, 10627, 7, 3571, 271, 84319, 7, 47059, 7, 47, 53, 23971, 11, 13, 5, 7, 201499, 5, 7, 67, 13, 7, 21211, 5, 29, 10696171, 11, 149, 971, 16896211, 11, 58111, 17, 11, 75307, 25105111, 853, 139, 7, 5, 613, 181, 23, 13, 29, 13, 19, 53, 47, 5, 11, 84811
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p(n), 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is in A094461;
6th-7th terms in A094462, A094463;

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=6};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

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

Views

Author

Labos Elemer, Oct 09 2000

Keywords

Comments

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

Examples

			a(3) = 11 is the smallest prime divisor of the form 6k + 5 of 6 * (5 * 29) - 1 = 6Q - 1 = 11 * 79 = 869.

References

  • 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.

Links

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A057204-A057208, A007528.

Programs

  • Mathematica
    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 *)
  • PARI
    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 */

Extensions

a(13)-a(17) from Robert Price, Jul 18 2015

A093779 a(n) is the position of prime 3 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

2, 1, 4, 3, 4, 3, 4, 3, 4, 4, 3, 3, 4, 3, 4, 4, 5, 3, 3, 4, 3, 3, 4, 4, 3, 5, 3, 4, 3, 4, 3, 4, 4, 3, 5, 3, 3, 3, 4, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5, 3, 3, 4, 3, 4, 3, 5, 3, 4, 3, 3, 4, 3, 5, 4, 3, 3, 3, 5, 5, 3, 4, 3, 4, 3, 4, 3, 3, 4, 4, 3, 5, 3, 4, 6, 3, 4, 3, 5, 4, 5, 3, 3, 3, 4, 5, 4, 3
Offset: 1

Views

Author

Labos Elemer, May 03 2004

Keywords

Examples

			p=3 arises first as n-th term for n=1,2,3,4 as follows: {3,2,7,43,13,53,5}, {2,3,7,43,13,53,5}, {7,2,3,43,13,53,5}, {5,2,11,3,331,19}, ... i.e., started at suitable initial primes;
p=2 arises always as 2nd or once as first term in case of various EM-sequences.

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777, A093778.

A093780 a(n) is the smallest prime used as initial value for Euclid-Mullin (EM) sequence (of variant A000945), such that in the corresponding EM-sequence the p=3 prime arises at the n-th position.

Original entry on oeis.org

3, 2, 7, 5, 59, 479, 821, 1871, 17393, 43019, 284783, 1572149, 2737793, 32938853, 24254639
Offset: 1

Views

Author

Labos Elemer, May 03 2004

Keywords

Examples

			p=3 arises first as n-th term for n=1,2,3,4,...,9th as follows:
{3,2,7,43,13,53,5},{2,3,7,43,13,53,5},{7,2,3,43,13,53,5},
{5,2,11,3,331,19},{269,2,7,3767,3,42559567},{479,2,7,19,5,3},
{821,2,31,109,7,509,3},{1871,2,19,7,37,13,23,3},
{17393,2,43,37,7,4129,13,5,3},

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777-A093779.

Extensions

More terms from David Wasserman, Apr 12 2007

A093781 a(n) is the position of the prime 5 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

7, 7, 1, 7, 18, 10, 3, 4, 11, 7, 8, 8, 10, 7, 3, 13, 8, 6, 7, 8, 6, 4, 7, 8, 9, 4, 6, 3, 4, 11, 5, 8, 3, 4, 4, 8, 8, 13, 3, 10, 21, 15, 6, 8, 3, 4, 13, 5, 3, 4, 8, 14, 6, 10, 3, 6, 12, 6, 10, 6, 6, 13, 8, 4, 6, 3, 11, 5, 3, 4, 13, 6, 10, 8, 4, 26, 8, 7, 11, 4, 7, 10, 7, 5, 4, 7, 16, 8, 7, 9, 3, 5, 5, 6
Offset: 1

Views

Author

Labos Elemer, May 04 2004

Keywords

Comments

a(38) = 13 because prime(38) = 163 and the corresponding EM sequence is {163, 2, 3, 11, 7, 75307, 3931, 5399, 3041, 409, 179, 92958641873, 5, 2003, ...}, where 5 appears at the 13th position. - David Wasserman, Apr 19 2007

Links

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777-A093780.
Cf. A093782.

Programs

  • PARI
    em(i) = local(p, c, n, f, q); p = prime(i); if (p == 5, return(1)); c = 1; n = p; while (1, c++; f = factor(n + 1, 2^31 - 1); q = f[1, 1]; if (!isprime(q), f = factor(n + 1); q = f[1, 1]); if (q == 5, return(c)); n *= q); \\ David Wasserman, Apr 19 2007

Extensions

More terms from David Wasserman, Apr 19 2007
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