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 1-10 of 10 results.

A143384 Duplicate of A139426.

Original entry on oeis.org

1, 5, 1, 5, 11, 11, 17, 19, 23, 97, 127, 145, 167, 269, 767, 479, 3307, 1453, 18007, 2357, 599, 17669, 5527, 3191, 3251, 70249, 147773, 39637
Offset: 1

Views

Author

Pierre CAMI, Aug 11 2008

Keywords

Comments

All primes certified using PFGW from Primeform group.

Examples

			(2^2-1)*(2^2-1+1)-1=11 prime 2^2-1=M(1) so a(1)=1
(2^3-1)*(2^3-1+1)-1=55 composite
(2^3-1)*(2^3-1+3)-1=69 composite
(2^3-1)*(2^3-1+5)-1=83 prime 2^3-1=M(2) so a(2)=5

Crossrefs

Cf. A000043 (Mersenne prime exponents), A143385.

Extensions

Missing terms 2357 and 599 inserted by Jens Kruse Andersen, Jul 30 2014

A139430 Smallest prime p such that M(n)^2+p*M(n)-1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

3, 5, 11, 5, 11, 11, 17, 19, 23, 97, 127, 1009, 167, 269, 953, 479, 3307, 1453, 37507, 2357, 599, 17669, 5527, 3191, 3251, 70249, 147773
Offset: 1

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			3*3+3*3-1=17 prime 3=M(1)=2^2-1 so p(1)=3;
7*7+5*7-1=83 prime 7=M(2)=2^3-1 so p(2)=5:
31*31+11*31-1=1301 prime 31=M(3)=2^5-1 so p(3)=11.

Crossrefs

Cf. A000668, A139424, A139425, A139426, A139427, A139428, A139429, A139431.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; p = 1;
While[! PrimeQ[m2 + Prime[p]*m - 1], p++]; Prime[p], {n, 18}] (* Robert Price, Apr 17 2019 *)

A139424 Smallest number k such that M(n)^2-k*M(n)-1 is prime with M(n) = Mersenne primes = A000668(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 43, 1, 41, 53, 91, 317, 317, 43, 1, 37, 3595, 563, 17239, 911, 11497, 58501, 1259, 10283, 138569, 72247, 27733, 11777, 179105
Offset: 1

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			3*3-1*3-1=5 prime 3=M(1)=2^2-1 so k(1)=1;
7*7-1*7-1=41 prime 7=M(2)=2^3-1 so k(2)=1;
31*31-1*31-1=929 prime 31=M(3)=2^5-1 so k(3)=1.

Crossrefs

Cf. A000668, A139425, A139426, A139427, A139428, A139429, A139430, A139421.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 - k*m - 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

Extensions

a(27)-a(28) from Robert Price, May 09 2019

A139425 Smallest number k such that M(n)^2-k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

1, 1, 9, 3, 3, 25, 7, 21, 435, 241, 3, 153, 151, 493, 537, 2871, 1713, 4941, 4963, 307, 28413, 5035, 1615, 43525, 9973
Offset: 1

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			3*3-1*3+1=7 prime 3=M(1)=2^2-1 so k(1)=1;
7*7-1*7+1=43 prime 7=M(2)=2^3-1 so k(2)=1;
31*31-9*31+1=683 prime 31=M(3)=2^5-1 so k(3)=9.

Crossrefs

Cf. A000668, A139424, A139426, A139427, A139428, A139429, A139430, A139421.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 - k*m + 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

A139427 Smallest number k such that M(n)^2+k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

1, 3, 5, 17, 17, 5, 83, 63, 71, 101, 543, 59, 569, 1029, 353, 1851, 2801, 2619, 525, 2907, 8955, 437, 30159, 5409, 8355
Offset: 1

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			3*3+1*3+1=13 prime 3=M(1)=2^2-1 so k(1)=1;
7*7+3*7+1=71 prime 7=M(2)=2^3-1 so k(2)=3;
31*31+5*31+1=1117 prime 31=M(3)=2^5-1 so k(3)=5.

Crossrefs

Cf. A000668, A139424, A139425, A139426, A139428, A139429, A139430, A139421.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 + k*m + 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

A139428 Smallest prime p such that M(n)^2-p*M(n)-1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

5, 7, 5, 17, 43, 67, 41, 53, 311, 317, 317, 43, 1427, 37, 25693, 563, 17239, 911, 11497, 112247, 1259, 190639, 138569, 296713, 27733, 11777
Offset: 2

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			7*7-5*7-1=13 prime 7=M(2)=2^3-1 so k(2)=5;
31*31-7*31-1=743 prime 31=M(3)=2^5-1 so k(3)=7.

Crossrefs

Cf. A000668, A139424, A139425, A139426, A139427, A139429, A139430, A139421.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; p = 1;
While[! PrimeQ[m2 - Prime[p]*m - 1], p++];
Prime[p], {n, 15}] (* Robert Price, Apr 17 2019 *)

A139429 Smallest prime p such that M(n)^2 - p*M(n) + 1 is prime with M(n) = A000668(n).

Original entry on oeis.org

3, 19, 3, 3, 73, 7, 271, 1021, 241, 3, 487, 151, 2971, 35839, 5737, 1723, 81943, 115741, 307, 151549, 231823, 443431, 195163, 9973, 114913, 362599
Offset: 2

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group.

Examples

			7*7-3*7+1=29 prime 7=M(2)=2^3-1 so k(2)=3;
31*31-19*31+1=373 prime 31=M(3)=2^5-1 so k(3)=19.

Crossrefs

Cf. A000668, A139424, A139425, A139426, A139427, A139428, A139430, A139421.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; p = 1;
While[! PrimeQ[m2 - Prime[p]*m + 1], p++];
Prime[p], {n, 15}] (* Robert Price, Apr 17 2019 *)

A143385 Least a(n) such that M(n)*(M(n)+a(n))-1 and M(n)*(M(n)+a(n))+1 are twin primes with M(i)=i-th Mersenne prime.

Original entry on oeis.org

1, 53, 11, 17, 317, 89, 737, 2543, 1247, 6209, 15107, 33119, 60611, 671063, 2648057
Offset: 1

Views

Author

Pierre CAMI, Aug 11 2008

Keywords

Examples

			(2^2-1)*(2^2-1+1)-1=11 prime, 11 and 13 twin primes, 2^2-1=M(1) so a(1)=1

Crossrefs

Cf. A139426.

A139431 Smallest prime p such that M(n)^2+p*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

3, 3, 5, 17, 17, 5, 83, 503, 71, 101, 947, 59, 569, 1787, 353, 17093, 2801, 3359, 25097, 9323, 491837, 97367, 209567, 21221, 273857, 462947, 216719
Offset: 1

Views

Author

Pierre CAMI, Apr 21 2008

Keywords

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			3*3+3*3+1=19 prime 3=M(1)=2^2-1 so p(1)=3;
7*7+3*7+1=71 prime 7=M(2)=2^3-1 so p(2)=3;
31*31+5*31+1=1117 prime 31=M(3)=2^5-1 so p(3)=5.

Crossrefs

Cf. A000668, A139424, A139425, A139426, A139427, A139428, A139429, A139430.

Programs

  • Mathematica
    A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; p = 1;
While[! PrimeQ[m2 + Prime[p]*m + 1], p++];
Prime[p], {n, 15}] (* Robert Price, Apr 17 2019 *)

A143387 Least prime a(n) such that M(n)*(M(n)+a(n))-1 and M(n)*(M(n)+a(n))+1 are twin primes with M(i)=i-th Mersenne prime A000043(i).

Original entry on oeis.org

3, 53, 11, 17, 317, 89, 1259, 2543, 7517, 16217, 15107, 33119, 60611, 671063, 2648057
Offset: 1

Views

Author

Pierre CAMI, Aug 11 2008

Keywords

Crossrefs

Cf. A000043, A139426, A143385, A143386.
Showing 1-10 of 10 results.

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