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.

A135977 Mersenne composites (A065341) with exactly 3 prime factors.

Original entry on oeis.org

536870911, 8796093022207, 140737488355327, 9007199254740991, 2361183241434822606847, 9444732965739290427391, 604462909807314587353087
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Links

Crossrefs

Subsequence of A065341.
Cf. A000225, A054723, A134852, A135975, A135976, A135978, A135979, A344515.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

Formula

a(n) = 2^A344515(n) - 1. - Amiram Eldar, May 23 2021

A135976 Mersenne composites (A065341) with exactly 2 prime factors.

Original entry on oeis.org

2047, 8388607, 137438953471, 2199023255551, 576460752303423487, 147573952589676412927, 9671406556917033397649407, 158456325028528675187087900671, 2535301200456458802993406410751
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Links

Crossrefs

Subsequence of A065341.
Cf. A000225, A054723, A134852, A135975, A135977, A135978, A135979.

Programs

  • Maple
    A135976 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) = 2) then
   RETURN (i)
fi: end: [ seq(A135976(n), n=1..26) ]; # Jani Melik, Feb 09 2011
  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k
  • PARI
    forprime(p=1, 1e2, if(bigomega(2^p-1)==2, print1(2^p-1, ", "))) \\ Felix Fröhlich, Aug 12 2014

Formula

a(n) = 2^A135978(n) - 1. - Amiram Eldar, May 23 2021

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

Original entry on oeis.org

5, 9, 12, 13, 17, 19, 23, 25, 26, 27, 29, 32, 33, 34, 35, 39, 45, 46, 49, 53, 57, 58, 60, 62, 69, 74, 75, 82, 88, 93, 99, 129, 140, 152, 164, 166, 168, 178, 179
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

a(40)>=206. - Amiram Eldar, Sep 29 2018

Crossrefs

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, n]]], {n, 1, 40}]; k
Select[Range[40],PrimeNu[2^Prime[#]-1]==2&] (* Harvey P. Dale, Jul 07 2013 *)

Formula

Equals {k: A001221(A001348(k)) = 2}. a(n) = A049084(A135978(n)). - R. J. Mathar, May 03 2008

Extensions

Edited by R. J. Mathar, May 03 2008
a(17)-a(34) from Donovan Johnson, Jun 14 2009
a(35)-a(39) from Amiram Eldar, Sep 29 2018

A089158 Second prime factor, if it exists, of Mersenne numbers.

Original entry on oeis.org

89, 178481, 1103, 616318177, 164511353, 9719, 4513, 69431, 3203431780337, 761838257287, 48544121, 2298041, 202029703, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393
Offset: 1

Views

Author

Cino Hilliard, Dec 06 2003

Keywords

Examples

			The 5th Mersenne number 2^11 - 1 = 23*89 and 89 is the second prime divisor.
The 9th Mersenne number 2^23 - 1 = 47*178481 and 178481 is the second prime divisor.
Notice 23, 89 congruent to 1 mod 11 and 47, 178481 congruent to 1 mod 23.

Links

Crossrefs

Cf. A001348, A016047, A065341, A089159, A089992, A135978.

Programs

  • PARI
    mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

A135980 Numbers k such that the Mersenne number 2^prime(k)-1 is composite.

Original entry on oeis.org

5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

A135979 is a subsequence of this sequence.

Crossrefs

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978, A135979.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], AppendTo[k, n]], {n, 1, 40}]; k
m = PrimePi @ MersennePrimeExponent @ Range[13]; Complement[Range[m[[-1]]], m] (* Amiram Eldar, Mar 12 2020 *)
  • PARI
    isok(k) = !isprime(2^prime(k)-1); \\ Michel Marcus, Mar 12 2020

Formula

prime(a(n)) = A054723(n).
a(n) = pi(A054723(n)).

Extensions

More terms from Amiram Eldar, Mar 12 2020

A344515 Primes p such that 2^p-1 has exactly 3 distinct prime factors.

Original entry on oeis.org

29, 43, 47, 53, 71, 73, 79, 179, 193, 211, 257, 277, 283, 311, 331, 349, 353, 389, 409, 443, 467, 499, 563, 577, 599, 613, 631, 643, 647, 683, 709, 751, 769, 829, 919, 941, 1039, 1103, 1117, 1123, 1171, 1193
Offset: 1

Views

Author

Amiram Eldar, May 21 2021

Keywords

Comments

The corresponding Mersenne numbers are in A135977.
a(43) >= 1237.
The following primes are also terms of this sequence: 1301, 1303, 1327, 1459, 1531, 1559, 1907, 2311, 2383, 2887, 3041, 3547, 3833, 4127, 4507, 4871, 6883, 7673, 8233.

Examples

			29 is a term since 2^29-1 = 536870911 = 233 * 1103 * 2089 has exactly 3 distinct prime factors.

Crossrefs

Subsequence of A054723.
Cf. A000225, A065341, A135975, A135976, A135977, A135978.

Programs

  • Mathematica
    Select[Range[200], PrimeQ[#] && PrimeNu[2^# - 1] == 3 &]

Formula

2^a(n) - 1 = A135977(n).

A292473 Square array read by antidiagonals downwards: A(n,k) = k-th prime p such that A001222(2^p-1) = n.

Original entry on oeis.org

2, 3, 11, 5, 23, 29, 7, 37, 43, 157, 13, 41, 47, 173, 113, 17, 59, 53, 181, 151, 223, 19, 67, 71, 229, 163, 239
Offset: 1

Views

Author

Felix Fröhlich, Sep 17 2017

Keywords

Comments

A permutation of the prime numbers.
Is this the same as k-th prime p such that A001221(2^p-1) = n?

Examples

			Array starts
    2,   3,   5,   7,  13,  17, ....
   11,  23,  37,  41,  59,  67, ....
   29,  43,  47,  53,  71,  73, ....
  157, 173, 181, 229, 233, 263, ....
  113, 151, 163, 191, 251, 307, ....
  223, 239, 359, 463, 587, 641, ....
  ....
A(2, 3) = 37, because the 3rd prime p such that 2^p-1 has 2 prime factors is 37, with 2^37-1 = 223 * 616318177.

Crossrefs

Cf. A000043 (row 1), A135978 (row 2), A140745 (column 1).
Cf. A001222, A088863.

Programs

  • Mathematica
    With[{s = Array[PrimeOmega[2^Prime@ # - 1] &, 50]}, Function[t, Function[u, Table[Prime@ u[[#, k]] &[n - k + 1], {n, Length@t}, {k, n, 1, -1}]]@ Map[PadRight[#, Length@ t] &, t]]@ Values@ KeySort@ PositionIndex@ s] // Flatten (* Michael De Vlieger, Sep 17 2017 *)

Extensions

More terms from Michael De Vlieger, Sep 17 2017
Showing 1-7 of 7 results.

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