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 1-4 of 4 results.

A122834 Primes in the new Mersenne conjecture; odd primes of the form 2^k+-1 or 4^k+-3.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 31, 61, 67, 127, 257, 1021, 4093, 4099, 8191, 16381, 65537, 65539, 131071, 262147, 524287, 1048573, 4194301, 16777213, 268435459, 1073741827, 2147483647, 2305843009213693951, 19342813113834066795298819
Offset: 1

Views

Author

T. D. Noe, Sep 12 2006

Keywords

Comments

Let p be a prime in this sequence. Call q=2^p-1 and r=(2^p+1)/3. The new Mersenne conjecture implies that either q and r are both prime or both composite.

References

  • R. K. Guy, Unsolved Problems in Number Theory, Section A3.

Links

Crossrefs

Superset of: A000668, A019434, A228026.
Cf. A000043 (n such that 2^n-1 is prime), A000978 (n such that (2^n+1)/3 is prime), A107360 (the intersection of these).

Programs

  • Mathematica
    nn=100; Union[Select[1+2^Range[16],PrimeQ], Select[ -1+2^Range[2nn],PrimeQ], Select[3+4^Range[nn],PrimeQ], Select[ -3+4^Range[nn],PrimeQ]]

A325204 Numbers k such that k*(k+1)*(k+2) has exactly 4 distinct prime factors.

Original entry on oeis.org

5, 9, 10, 11, 12, 14, 15, 17, 18, 22, 23, 24, 25, 26, 27, 30, 31, 32, 36, 46, 47, 48, 52, 62, 71, 72, 79, 80, 81, 96, 106, 107, 126, 127, 162, 191, 192, 241, 242, 256, 382, 431, 486, 512, 576, 862, 1151, 1152, 2186, 2591, 2592, 2916, 4372, 8191, 8746, 131071, 131072, 139967, 472391, 524287, 786431, 995326, 995327
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Sep 05 2019

Keywords

Comments

Contains 2^p-1 for p in A107360 except 3.
Contains all terms of A325255 except 2 and 4.
Contains k-1 for k in A027856 except 4.
Contains k-2 for k in A327240 except 6 and 8. - Ray Chandler, Sep 14 2019

Examples

			a(3)=10 is in the sequence because 10*11*12 has four distinct prime factors: 2, 3, 5, 11.

Links

Crossrefs

Cf. A027856, A107360, A325255, A327240.

Programs

  • Maple
    select(t -> nops(numtheory:-factorset(t) union numtheory:-factorset(t+1) union numtheory:-factorset(t+2))=4, [$1..10^6]);
  • PARI
    select(k->4==omega(k*(k+1)*(k+2)), [1..10000]) \\ Andrew Howroyd, Sep 05 2019

A109799 Primes p such that 2^p - 1 is a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 61, 127
Offset: 1

Views

Author

Jason Earls, Aug 15 2005

Keywords

Comments

For p in this sequence, 2^p - 1 is called a Mersenne-Chen prime.
Conjecture: 2^127 - 1 is the largest Mersenne-Chen prime.
Except for the initial term 2, this sequence is the intersection of A000043 and A000978 given by A107360. - Max Alekseyev, Oct 28 2008, Jan 28 2010

Examples

			a(5)=13 because 2^13 - 1 = 8191 is prime and 2^13 + 1 = 3*2731 is semiprime.

Crossrefs

Cf. A000043, A000978, A107360, A109611.

Programs

  • Mathematica
    Select[Prime[Range[40]],PrimeQ[2^#-1]&&PrimeOmega[2^#+1]<3&] (* James C. McMahon, Mar 30 2024 *)

A302333 Wagstaff primes related to The New Mersenne Conjecture that are the indices of perfect numbers in a list of centered 9-gonal numbers.

Original entry on oeis.org

3, 11, 43, 2731, 43691, 174763, 715827883, 768614336404564651, 56713727820156410577229101238628035243
Offset: 1

Views

Author

Steve Homewood, Apr 05 2018

Keywords

Comments

Let p be a Wagstaff prime related to The New Mersenne Conjecture. Then (3p-2)(3p-1)/2 gives the perfect number whose index it is.

Examples

			For p = 3, (3*3-2)*(3*3-1)/2 = 28 and for p = 11, (3*11-2)(3*11-1)/2 = 496.

Links

Crossrefs

Cf. A000979, A107360.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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