A135980 -id:A135980 - cp's OEIS frontend

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

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

Offset: 1

Labos Elemer, Oct 30 2001

nonn

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

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

Muniru A Asiru, Table of n, a(n) for n = 1..110
John H. Jaroma and Kamaliya N. Reddy, Classical and alternative approaches to the Mersenne and Fermat numbers, The American Mathematical Monthly, Vol. 114, No. 8 (2007), pp. 677-687.

Cf. A054723, A000043, A000668, A001348, A001567.

Cf. A135975, A135980, A145097, A136031. - Artur Jasinski, Oct 01 2008

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

Select[Table[2^Prime[n]-1,{n,30}],!PrimeQ[#]&] (* Harvey P. Dale, May 06 2018 *)

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

Artur Jasinski, Oct 01 2008

nonn

Note that not all the largest divisors are primes.

Which divisors are prime? - see A145099. - Artur Jasinski, Oct 04 2008

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A065341, A135975, A135980, A136031.

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

Name clarified by Amiram Eldar, Mar 12 2020

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

Artur Jasinski, Oct 01 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..191

Cf. A065341, A135975, A135980, A145097, A136031.

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

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

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

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

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

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