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 1-8 of 8 results.

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

A135978 Primes p such that 2^p-1 has exactly 2 prime factors.

Original entry on oeis.org

11, 23, 37, 41, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997, 1061, 1063
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

a(40)>=1277. - Amiram Eldar, Sep 29 2018

Crossrefs

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

Programs

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

Extensions

a(17)-a(37) from Arkadiusz Wesolowski, Jan 26 2012
a(38)-a(39) from Amiram Eldar, Sep 29 2018

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

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

A089159 If Mersenne numbers have 3 or more factors, then list the third factor.

Original entry on oeis.org

2089, 2099863, 13264529, 20394401, 212885833, 9361973132609, 1113491139767, 65993, 165799, 1654058017289, 110211473, 70084436712553223, 1489459109360039866456940197095433721664951999121, 7648337, 39940132241, 14732265321145317331353282383
Offset: 1

Views

Author

Cino Hilliard, Dec 06 2003, corrected Nov 16 2006

Keywords

Examples

			The 10th Mersenne number 2^29 - 1 = 233*1103*2089 and 2089 is the third prime factor. Notice these factors are congruent to 1 (mod 29).

Links

Crossrefs

Cf. A000043, A001348, A016047, A089158, A135977, A344515.

Programs

  • PARI
    mersenne2(n) = { c=0; forprime(x=2, n, c++; y = 2^x-1; f=ifactor(y); if(length(f)>=3, print1(f[3]","); ) ) }
ifactor(n) = { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

Formula

A Mersenne number (A001348) is a number of the form 2^p - 1 where p is prime.

Extensions

a(16) from Amiram Eldar, Jul 11 2024

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).

A135386 Mersenne composites A065341 with 4 or more prime factors.

Original entry on oeis.org

10384593717069655257060992658440191, 2854495385411919762116571938898990272765493247, 182687704666362864775460604089535377456991567871
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Links

Crossrefs

Subsequence of A065341.
Cf. A000225, A054723, A134852, A135975, A135976, A135977, A135979.

Programs

  • Maple
    A135386 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) > 3) then
   RETURN (i)
fi: end: seq(A135386(n), n=1..37); # 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 >3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

A294729 Numbers n such that 2^n - 1 is the product of three primes.

Original entry on oeis.org

6, 8, 10, 14, 15, 25, 26, 27, 29, 34, 38, 43, 47, 53, 62, 65, 71, 73, 79, 85, 93, 122, 133, 179, 193, 211, 254, 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
Offset: 1

Views

Author

Arkadiusz Wesolowski, Nov 07 2017

Keywords

Comments

The eighteenth composite term is 3481. No other composite terms up to 10000.

Examples

			a(1) = 6 because 2^6 - 1 = 63 = 3^2*7 is a 3-almost prime.
a(2) = 8 because 2^8 - 1 = 255 = 3*5*17 is a 3-almost prime.

Links

Crossrefs

Cf. A000043 (product of one prime), A000225, A085724 (product of two primes), A135977.

Programs

  • Magma
    lst:=[]; factors:=func; bigomega:=func; IsCube:=func; for n in [2..254] do if IsPrime(n) then if bigomega(2^n-1) eq 3 then Append(~lst, n); end if; else f:=factors(n); a:=f[1,1]; if IsPrime(2^a-1) then if IsSquarefree(n) then if bigomega(n) eq 2 then b:=f[2,1]; if IsPrime(2^b-1) and IsPrime(Truncate((2^n-1)/((2^a-1)*(2^b-1)))) then Append(~lst, n); end if; end if; end if; if IsSquare(n) or IsCube(n) then if bigomega(Truncate((2^n-1)/(2^a-1))) eq 2 then Append(~lst, n); end if; end if; end if; end if; end for; lst;
  • Mathematica
    ParallelMap[ If[ PrimeOmega[2^# - 1] == 3, #, Nothing] &, Range@1250] (* Robert G. Wilson v, Nov 28 2017 *)
  • PARI
    is(n)=bigomega(2^n-1)==3

Extensions

a(28)-a(56) added from the Cunningham project
Showing 1-8 of 8 results.

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

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