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 1-6 of 6 results.

A145098 Duplicate of A136031.

Original entry on oeis.org

89, 178481, 2089, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 761838257287, 212885833, 9361973132609
Offset: 1

Views

Author

Keywords

A065341 Mersenne composites: 2^prime(m) - 1 is not a prime.

Original entry on oeis.org

2047, 8388607, 536870911, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 147573952589676412927, 2361183241434822606847, 9444732965739290427391
Offset: 1

Views

Author

Labos Elemer, Oct 30 2001

Keywords

Comments

For the number of prime factors in a(n) see A135975. For indices of primes n in composite 2^prime(n)-1 see A135980. For smallest prime divisors of Mersenne composites see A136030. For largest prime divisors of Mersenne composites see A136031. For largest divisors see A145097. - Artur Jasinski, Oct 01 2008
All the terms are Fermat pseudoprimes to base 2 (A001567). For a proof see, e.g., Jaroma and Reddy (2007). - Amiram Eldar, Jul 24 2021

Examples

			2^11 - 1 = 2047 = 23*89.
		

Crossrefs

Programs

  • Maple
    A065341 := proc(n) local i;
    i := 2^(ithprime(n))-1:
    if (not isprime(i)) then
       RETURN (i)
    fi: end: seq(A065341(n), n=1..21); # Jani Melik, Feb 09 2011
  • Mathematica
    Select[Table[2^Prime[n]-1,{n,30}],!PrimeQ[#]&] (* Harvey P. Dale, May 06 2018 *)

Formula

a(n) = 2^A054723(n) - 1.

A145097 a(n) is the largest proper divisor of the Mersenne composite A065341(n).

Original entry on oeis.org

89, 178481, 2304167, 616318177, 164511353, 20408568497, 59862819377, 1416003655831, 3203431780337, 761838257287, 10334355636337793, 21514198099633918969, 224958284260258499201, 57912614113275649087721
Offset: 1

Views

Author

Artur Jasinski, Oct 01 2008

Keywords

Comments

Note that not all the largest divisors are primes.
Which divisors are prime? - see A145099. - Artur Jasinski, Oct 04 2008

Crossrefs

Programs

  • Mathematica
    a = {}; Do[m = 2^Prime[n] - 1; If[PrimeQ[m], null, AppendTo[a, Divisors[m][[ -2]]]], {n, 1, 40}]; a

Extensions

Name clarified by Amiram Eldar, Mar 12 2020

A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

Original entry on oeis.org

2, 4, 6, 16, 12, 18, 24, 40, 54, 36, 102, 110, 60, 72, 108, 140, 120, 156, 144, 200, 216, 210, 240, 180, 456, 288, 336, 300, 396, 480, 882, 360, 468, 700
Offset: 1

Views

Author

Artur Jasinski, Dec 11 2007

Keywords

Crossrefs

Programs

  • Maple
    N:= 24: # to get a(1) to a(N)
    unknown:= N:
    for k from 2 while unknown > 0 do
      q:= numtheory:-bigomega(2^k-1);
      if q <= N and not assigned(A[q]) then
         A[q]:= k;
         unknown:= unknown - 1;
      fi
    od:
    seq(A[i],i=1..N); # Robert Israel, Oct 24 2014
  • Mathematica
    Module[{nn=250,tbl},tbl=Table[{k,PrimeOmega[2^k-1]},{k,nn}];Table[SelectFirst[tbl,#[[2]]==n&],{n,24}]][[;;,1]] (* The program generates the first 24 terms of the sequence. *)  (* Harvey P. Dale, May 25 2025 *)
  • PARI
    a(n) = {k = 1; while(bigomega(2^k-1) != n, k++); k;} \\ Michel Marcus, Nov 04 2013

Extensions

a(15)-a(20) from Michel Marcus, Nov 04 2013
a(21)-a(24) from Derek Orr, Oct 23 2014
a(25)-a(34) from Jinyuan Wang, Jun 07 2019

A145099 a(n) = 1 if the largest proper divisor of Mersenne composite A145097(n) is prime and a(n) = 0 in opposite case.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
Offset: 1

Views

Author

Artur Jasinski, Oct 01 2008

Keywords

Crossrefs

Programs

  • Mathematica
    a = {}; Do[m = 2^Prime[n] - 1; k = Divisors[m][[ -2]]; If[PrimeQ[m], null, If[PrimeQ[k], AppendTo[a, 1], AppendTo[a, 0]]], {n, 1, 50}]; a

Extensions

Name clarified and more terms added by Amiram Eldar, Mar 12 2020

A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 24, 40, 36, 48, 88, 60, 72, 150, 132, 120, 156, 144, 200, 204, 210, 180, 324, 476, 288, 300, 432, 396, 480, 360, 468, 576, 700, 504, 420, 648, 540, 660, 792, 720
Offset: 0

Views

Author

Artur Jasinski, Dec 11 2007

Keywords

Comments

First occurrence of n in A046800.

Crossrefs

Programs

  • Mathematica
    With[{pn1=PrimeNu[2^Range[800]-1]},Table[Position[pn1,n,1,1],{n,0,40}]]//Flatten (* Harvey P. Dale, Jan 10 2025 *)
  • PARI
    a(n) = my(k=1); while (omega(2^k-1) != n, k++); k; \\ Michel Marcus, Jan 09 2023

Extensions

More terms from Julián Aguirre, Feb 04 2013
a(31)-a(39) from Chai Wah Wu, Oct 03 2019
a(0) = 1 inserted by Michel Marcus, Jan 09 2023
Showing 1-6 of 6 results.