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 1-7 of 7 results.

A183071 Positive integers k such that each prime divisor of 2^k - 1 has the form 4j + 3.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 26, 30, 31, 34, 38, 43, 51, 61, 62, 65, 79, 85, 86, 89, 93, 95, 107, 122, 127, 129, 130, 133, 158, 170, 193, 254, 255, 301, 311, 331, 349, 389, 445, 521, 557, 577, 579, 607, 631, 647, 1103, 1167
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

The exponents of the Mersenne primes (A000043) are contained in this sequence.
Needed factorizations are in the Cunningham Project.

Examples

			6 is in this sequence because 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

Links

Crossrefs

Cf. A000043, A136003, A183072, A183073, A183074.
Cf. A000668, A136005, A183075, A183076, A183077, A183078.

Formula

A183075(n) = 2^a(n) - 1.

Extensions

a(53)-a(54) from Amiram Eldar, Feb 18 2019

A183076 Composite numbers of the form 2^k - 1 for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

63, 1023, 16383, 32767, 67108863, 1073741823, 17179869183, 274877906943, 8796093022207, 2251799813685247, 4611686018427387903, 36893488147419103231
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

Needed factorizations are in the Cunningham Project.

Examples

			63 = 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

Links

Crossrefs

Cf. A000668, A136005, A183075, A183077, A183078.
Cf. A000043, A136003, A183071, A183072, A183073, A183074.

Formula

a(n) = 2^A183072(n) - 1.

A183077 Numbers of the form 2^p - 1, with p prime, for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 8796093022207, 2305843009213693951, 604462909807314587353087, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

The Mersenne primes (A000668) are contained in this sequence.
Needed factorizations are in the Cunningham Project.

Examples

			8796093022207 = 2^43 - 1 = 431 * 9719 * 2099863.

Links

Crossrefs

Cf. A000668, A136005, A183075, A183076, A183078.
Cf. A000043, A136003, A183071, A183072, A183073, A183074.

Formula

a(n) = 2^A183073(n) - 1.

A183078 Composite numbers of the form 2^p - 1, with p prime, for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

8796093022207, 604462909807314587353087, 12554203470773361527671578846415332832204710888928069025791, 4171849679533027504677776769862406473833407270227837441302815640277772901915313574263597826047
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

Needed factorizations are in the Cunningham Project.

Examples

			8796093022207 = 2^43 - 1 = 431 * 9719 * 2099863.

Links

Crossrefs

Cf. A000668, A136005, A183075, A183076, A183077.
Cf. A000043, A136003, A183071, A183072, A183073, A183074.

Formula

a(n) = 2^A183074(n) - 1.

A183072 Positive integers k such that 2^k - 1 is composite and each of its prime divisors has the form 4j + 3.

Original entry on oeis.org

6, 10, 14, 15, 26, 30, 34, 38, 43, 51, 62, 65, 79, 85, 86, 93, 95, 122, 129, 130, 133, 158, 170, 193, 254, 255, 301, 311, 331, 349, 389, 445, 557, 577, 579, 631, 647, 1103, 1167
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

Needed factorizations are in the Cunningham Project.

Examples

			6 is in this sequence because 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

Links

Crossrefs

Cf. A000043, A136003, A183071, A183073, A183074.
Cf. A000668, A136005, A183075, A183076, A183077, A183078.

Programs

  • Mathematica
    c4j3Q[n_]:=Module[{c=2^n-1},CompositeQ[c]&&AllTrue[(#-3)/4&/@ FactorInteger[ c] [[All,1]],IntegerQ]]; Select[Range[650],c4j3Q] (* Requires Mathematica version 10 or later *) (* The program takes a long time to run *) (* Harvey P. Dale, Sep 23 2018 *)

Formula

A183076(n) = 2^a(n) - 1.

Extensions

a(38)-a(39) from Amiram Eldar, Feb 18 2019

A183073 Prime numbers p such that each prime divisor of 2^p - 1 has the form 4j + 3.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 43, 61, 79, 89, 107, 127, 193, 311, 331, 349, 389, 521, 557, 577, 607, 631, 647, 1103
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

The exponents of the Mersenne primes (A000043) are contained in this sequence.
Needed factorizations are in the Cunningham Project.
Also in the sequence are 1279, 2203, 2281, 2909, 3217, 4253. - Amiram Eldar, Feb 18 2019

Examples

			43 is in this sequence because 2^43 - 1 = 431 * 9719 * 2099863, and each of those primes has the form 4j + 3.

Links

Crossrefs

Cf. A000043, A136003, A183071, A183072, A183074.
Cf. A000668, A136005, A183075, A183076, A183077, A183078.

Programs

  • Mathematica
    Select[Prime[Range[30]],And@@(IntegerQ[(#-3)/4]&/@Transpose[ FactorInteger[ 2^#-1]][[1]])&] (* Increase the value of Range to increase the number of terms generated, but processing times grow very quickly as the value increases. *)(* Harvey P. Dale, Jan 01 2013 *)

Formula

A183077(n) = 2^a(n) - 1.

Extensions

a(26) from Amiram Eldar, Feb 18 2019

A183074 Prime numbers p such that 2^p - 1 is composite and each of its prime divisors has the form 4j + 3.

Original entry on oeis.org

43, 79, 193, 311, 331, 349, 389, 557, 577, 631, 647, 1103
Offset: 1

Views

Author

Stuart Clary, Dec 23 2010

Keywords

Comments

Needed factorizations are in the Cunningham Project.

Examples

			43 is in this sequence because 2^43 - 1 = 431 * 9719 * 2099863, and each of those primes has the form 4j + 3.

Links

Crossrefs

Cf. A000043, A136003, A183071, A183072, A183073.
Cf. A000668, A136005, A183075, A183076, A183077, A183078.

Programs

  • Mathematica
    cQ[n_]:=Module[{x=2^n-1},!PrimeQ[x]&&Union[Mod[Transpose[ FactorInteger[ x]][[1]],4]]=={3}]; Select[Prime[Range[120]],cQ] (* Harvey P. Dale, Jun 17 2014 *)

Formula

A183078(n) = 2^a(n) - 1.

Extensions

a(12) from Amiram Eldar, Feb 18 2019
Showing 1-7 of 7 results.

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