A135975 -id:A135975 - 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.

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

Original entry on oeis.org

536870911, 8796093022207, 140737488355327, 9007199254740991, 2361183241434822606847, 9444732965739290427391, 604462909807314587353087

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..42

Subsequence of A065341.

Cf. A000225, A054723, A134852, A135975, A135976, A135978, A135979, A344515.

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

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

Artur Jasinski, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..39
Wikipedia, Semiprime.

Subsequence of A065341.

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

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

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

forprime(p=1, 1e2, if(bigomega(2^p-1)==2, print1(2^p-1, ", "))) \\ Felix Fröhlich, Aug 12 2014

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

Artur Jasinski, Dec 09 2007

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

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

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

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

Artur Jasinski, Dec 09 2007

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

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

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 *)

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

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

Artur Jasinski, Dec 09 2007

nonn

A135979 is a subsequence of this sequence.

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

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 *)

isok(k) = !isprime(2^prime(k)-1); \\ Michel Marcus, Mar 12 2020

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

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

More terms from Amiram Eldar, Mar 12 2020

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

A136032 Number of prime factors (with multiplicity) of Mersenne composites (A065341).

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 5, 4, 5, 2, 4, 3, 4, 5, 3, 2, 2, 3, 6, 2, 4, 4, 6, 2, 5, 3, 4, 2, 2, 3, 2, 3, 2, 5, 3, 4, 4, 3, 5, 2, 3, 3, 6, 5, 2, 2, 5, 3, 9, 4, 3, 5, 2, 8, 4, 4, 3, 5, 2, 4, 6, 3, 4, 2, 7, 3, 4, 4, 2, 5, 4, 5, 3, 5, 4

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

If the conjecture that all Mersenne composites are squarefree is true, then this sequence is identical to A135975. - Felix Fröhlich, Aug 24 2014

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

Cf. A001222, A065341, A135975.

a = {}; Do[If[PrimeQ[n] && !PrimeQ[2^n - 1], w = 2^n - 1; c = FactorInteger[w]; d = Length[c]; b = 0; Do[b = b + c[[k]][[2]], {k, 1, d}]; AppendTo[a, b]], {n, 2, 150}]; a
PrimeOmega/@Select[2^Prime[Range[100]]-1,!PrimeQ[#]&] (* Harvey P. Dale, Nov 01 2016 *)

forprime(p=2, 1e3, if(!ispseudoprime(2^p-1), print1(bigomega(2^p-1), ", "))) \\ Felix Fröhlich, Aug 24 2014

a(n) = A001222(A065341(n)). - Michel Marcus, Aug 24 2014

More terms from Michel Marcus, Nov 04 2013
Definition adjusted by Felix Fröhlich, Aug 24 2014
More terms from Felix Fröhlich, Aug 24 2014

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

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

Amiram Eldar, May 21 2021

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.

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

Subsequence of A054723.

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

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

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

cp's OEIS Frontend

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

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

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

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

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

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

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

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A136032 Number of prime factors (with multiplicity) of Mersenne composites (A065341).

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