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.

Showing 1-6 of 6 results.

A093777 a(n) is the smallest prime which, if used to start a Euclid-Mullin sequence (like A000945), the resulting sequence contains the n consecutive primes 2, 3, ..., prime(n).

Original entry on oeis.org

2, 2, 19, 199, 2089, 99109, 1960969, 10129129, 87726649, 4549584049, 328034245549, 20584643748679, 666188861477149, 31395465477725359, 894857713367947339, 434392154438254391389, 17934770256689308411399
Offset: 1

Views

Author

Labos Elemer, May 03 2004

Keywords

Comments

Thanks in part to Dirichlet's theorem, a(n) exists for each n. - Don Reble, Oct 07 2006

Examples

			a(1) = a(2) = 2 because they generate {2,3,7,43,13,...};
a(3) = 19 because it generates {19,2,3,5,571,271,...}, see A051312;
a(4) = 199 because it generates {199,2,3,5,7,23,881,...};
a(5) = 2089 because it generates {2089,2,3,5,7,11,269,...};
a(6) = 99109 because it generates {99109,2,3,5,7,11,13,2976243271,...};
a(7) = 1960969 because it generates {1960969,2,3,5,7,11,13,17,281,47,419,5539788476533581271,37,19,173,...}

Links

Crossrefs

Cf. A093778, A093779, A093780, A000945, A051309-A051334, A051614-A051616, A056756.

Extensions

More terms from Don Reble, Oct 07 2006

A094460 a(n) is the third term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

7, 7, 11, 3, 23, 3, 5, 3, 47, 59, 3, 3, 83, 3, 5, 107, 7, 3, 3, 11, 3, 3, 167, 179, 3, 7, 3, 5, 3, 227, 3, 263, 5, 3, 13, 3, 3, 3, 5, 347, 359, 3, 383, 3, 5, 3, 3, 3, 5, 3, 467, 479, 3, 503, 5, 17, 7, 3, 3, 563, 3, 587, 3, 7, 3, 5, 3, 3, 5, 3, 7, 719, 3, 3, 3, 13, 19, 3, 11, 3, 839, 3, 863
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p[n], 2nd equals 2; 3rd term is given here as largest p-divisor of 2p+1 [occasionally safe primes, A005385];
4th terms listed in A051614; further terms are in A094461-A094463.

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308 - A051335, A093778-A093782, A094152/A094153.
Except for first term [which is A000945(3)], the same as A023592.

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=10}; Print[a[el]; ta[[j]]=a[el]; j++ ], {j, 1, 168}]; ta

Formula

a(n)= a(n-1)+ A008472(a(n-1)) - Ctibor O. Zizka, May 26 2008

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

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.
Showing 1-6 of 6 results.

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