A000668 -id:A000668 - cp's OEIS frontend

A248934 Decimal expansion of 2^3217 - 1, the 18th Mersenne prime A000668(18).

2, 5, 9, 1, 1, 7, 0, 8, 6, 0, 1, 3, 2, 0, 2, 6, 2, 7, 7, 7, 6, 2, 4, 6, 7, 6, 7, 9, 2, 2, 4, 4, 1, 5, 3, 0, 9, 4, 1, 8, 1, 8, 8, 8, 7, 5, 5, 3, 1, 2, 5, 4, 2, 7, 3, 0, 3, 9, 7, 4, 9, 2, 3, 1, 6, 1, 8, 7, 4, 0, 1, 9, 2, 6, 6, 5, 8, 6, 3, 6, 2, 0, 8, 6, 2, 0, 1, 2, 0, 9, 5, 1, 6, 8, 0, 0, 4, 8, 3, 4, 0, 6, 5, 5, 0

Offset: 969

Arkadiusz Wesolowski, Oct 17 2014

The prime was found on September 8, 1957, by Hans Riesel, using BESK.

			25911708601320262777624676792244153094181888755312542730397492316187401...

Arkadiusz Wesolowski, Table of n, a(n) for n = 969..1937
Hans Riesel, A New Mersenne Prime, Mathematics of Computation, vol. 12 (1958), p. 60.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000668(16), A248933 = A000668(17), A248935 = A000668(19), A248936 = A000668(20).

Magma
```
Reverse(Intseq(2^3217-1));
```

RealDigits[2^3217 - 1, 10, 100][[1]] (* G. C. Greubel, Oct 03 2017 *)

PARI
```
eval(Vec(Str(2^3217-1)))
```

Equals 2^A000043(18) - 1.

A248935 Decimal expansion of 2^4253 - 1, the 19th Mersenne prime A000668(19).

Original entry on oeis.org

1, 9, 0, 7, 9, 7, 0, 0, 7, 5, 2, 4, 4, 3, 9, 0, 7, 3, 8, 0, 7, 4, 6, 8, 0, 4, 2, 9, 6, 9, 5, 2, 9, 1, 7, 3, 6, 6, 9, 3, 5, 6, 9, 9, 4, 7, 4, 9, 9, 4, 0, 1, 7, 7, 3, 9, 4, 7, 4, 1, 8, 8, 2, 6, 7, 3, 5, 2, 8, 9, 7, 9, 7, 8, 7, 0, 0, 5, 0, 5, 3, 7, 0, 6, 3, 6, 8, 0, 4, 9, 8, 3, 5, 5, 1, 4, 9, 0, 0, 2, 4, 4, 3, 0, 3

Offset: 1281

Arkadiusz Wesolowski, Oct 17 2014

This prime and the 20th Mersenne prime were found in 1961 by Alexander Hurwitz, using IBM 7090.

			19079700752443907380746804296952917366935699474994017739474188267352897...

Arkadiusz Wesolowski, Table of n, a(n) for n = 1281..2561
Alexander Hurwitz, New Mersenne Primes, Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249-251.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000668(16), A248933 = A000668(17), A248934 = A000668(18), A248936 = A000668(20).

Magma
```
Reverse(Intseq(2^4253-1));
```

RealDigits[2^4253 - 1, 10, 100][[1]] (* G. C. Greubel, Oct 03 2017 *)

PARI
```
eval(Vec(Str(2^4253-1)))
```

Equals 2^A000043(19) - 1.

A248936 Decimal expansion of 2^4423 - 1, the 20th Mersenne prime A000668(20).

Original entry on oeis.org

2, 8, 5, 5, 4, 2, 5, 4, 2, 2, 2, 8, 2, 7, 9, 6, 1, 3, 9, 0, 1, 5, 6, 3, 5, 6, 6, 1, 0, 2, 1, 6, 4, 0, 0, 8, 3, 2, 6, 1, 6, 4, 2, 3, 8, 6, 4, 4, 7, 0, 2, 8, 8, 9, 1, 9, 9, 2, 4, 7, 4, 5, 6, 6, 0, 2, 2, 8, 4, 4, 0, 0, 3, 9, 0, 6, 0, 0, 6, 5, 3, 8, 7, 5, 9, 5, 4, 5, 7, 1, 5, 0, 5, 5, 3, 9, 8, 4, 3, 2, 3, 9, 7, 5, 4

Offset: 1332

Arkadiusz Wesolowski, Oct 17 2014

The 19th Mersenne prime and this prime were found in 1961 by Alexander Hurwitz, using IBM 7090.

			28554254222827961390156356610216400832616423864470288919924745660228440...

Arkadiusz Wesolowski, Table of n, a(n) for n = 1332..2663
Alexander Hurwitz, New Mersenne Primes, Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249-251.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000668(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19).

Magma
```
Reverse(Intseq(2^4423-1));
```

RealDigits[2^4423 - 1, 10, 100][[1]] (* G. C. Greubel, Oct 03 2017 *)

PARI
```
eval(Vec(Str(2^4423-1)))
```

Equals 2^A000043(20) - 1.

A133049 Squares of Mersenne primes A000668(n).

Original entry on oeis.org

9, 49, 961, 16129, 67092481, 17179607041, 274876858369, 4611686014132420609, 5316911983139663487003542222693990401, 383123885216472214589586755549637256619304505646776321

Offset: 1

Omar E. Pol, Oct 30 2007, Apr 23 2008

nonn

Sum of last A000043(n) divisors of the n-th even perfect number. In other words; sum of divisors that are not powers of 2 of the n-th even perfect number, or sum of divisors that are multiples of the n-th Mersenne prime A000668(n) of the n-th even perfect number. See A139247 for more information.

See the structure of the divisors of perfect numbers in A135652, A135653, A135654 and A135655.

			a(3)=961 because the 3rd Mersenne prime is 31 and 31^2=961.

G. C. Greubel, Table of n, a(n) for n = 1..15
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000290, A001248. Mersenne primes: A000668.

Cf. A000043, A000396, A135652, A135653, A135654, A135655, A138247.

Select[2^Range[1000] - 1, PrimeQ]^2 (* G. C. Greubel, Oct 03 2017 *)

forprime(p=2, 1000, if(ispseudoprime(2^p-1), print1((2^p-1)^2", "))) \\ G. C. Greubel, Oct 03 2017

a(n) = A000668(n)^2

More terms from Olaf Voß, Feb 13 2008

A138837 Non-Mersenne primes: A000040 \ A000668.

Original entry on oeis.org

2, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Omar E. Pol, Apr 03 2008

Primes that are not Mersenne primes A000668.
Pandigital primes in base 2. (Pandigital interpreted as including all digits, not necessarily only once each.) - Franklin T. Adams-Watters, May 11 2011
Primes whose sum of divisors is not a power of 2. - Omar E. Pol, Dec 19 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Microsoft Ltd., The Prime Challenge, Dec. 2013
OEIS wiki, Pandigital primes.

Primes in A133398.

Cf. A000040, A000203, A000668, A155151.

max = 300; Complement[Prime[Range[PrimePi[max]]], 2^Range[Ceiling[Log[2, max]]] - 1] (* Alonso del Arte, Dec 30 2013 *)

is_A138837(n)={isprime(n)&&1<M. F. Hasler, Feb 05 2014

A138837 = A000040 \ A000668. - M. F. Hasler, Feb 09 2014

New name from Omar E. Pol, Jan 02 2014

A139295 a(n) = 2^(2p - 1)-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

31, 8191, 2305843009213693951, 14474011154664524427946373126085988481658748083205070504932198000989141204991

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A000668, A139294, A139296, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 3) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

a(n) = 2^(2*A000668(n)-1)-1 = A139294(n)-1.

A139231 First differences of Mersenne primes A000668.

Original entry on oeis.org

4, 24, 96, 8064, 122880, 393216, 2146959360, 2305843007066210304, 618970017336847128235868160, 162258657859193720701440560726016, 170141021201192402518323912137873817600

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

			a(1)=4 because A000668(1)=3 and A000668(2)=7 then 7-3 = 4.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000668, A139228, A139229, A139230, A139232, A139233, A139234, A139235, A139236, A139237.

A000668 := Select[2^Range[1000] - 1, PrimeQ]; Table[A000668[[n + 1]] - A000668[[n]], {n, 1, 10}] (* G. C. Greubel, Oct 03 2017 *)
Differences[2^MersennePrimeExponent[Range[20]]-1] (* Harvey P. Dale, Mar 31 2022 *)

a=0; b=0; forprime(p=1, 1e2, if(ispseudoprime(2^p-1) && a==0, a=2^p-1); if(ispseudoprime(2^p-1) && a!=0, b=2^p-1; if(a!=b, print1(b-a, ", ")); a=b)) \\ Felix Fröhlich, Aug 12 2014

a(n) = A000668(n+1) - A000668(n).

a(8)-a(11) from Felix Fröhlich, Aug 12 2014

A139232 Second differences of Mersenne primes A000668.

Original entry on oeis.org

20, 72, 7968, 114816, 270336, 2146566144, 2305843004919250944, 618970015031004121169657856, 162258038889176383854312324857856, 170140858942534543324603210697313091584

Offset: 1

Omar E. Pol, Apr 19 2008

nonn

Second differences of even superperfect numbers, multiplied by 2 (see A139236).

G. C. Greubel, Table of n, a(n) for n = 1..16

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139233, A139234, A139235, A139236, A139237.

A000668 := Select[2^Range[1000] - 1, PrimeQ]; Table[A000668[[n + 2]] - 2*A000668[[n + 1]] + A000668[[n]], {n, 1, 6}] (* G. C. Greubel, Oct 03 2017 *)
Differences[2^MersennePrimeExponent[Range[14]]-1,2] (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139236(n)*2.

Terms a(7) - a(10) added by G. C. Greubel, Oct 03 2017

A139296 a(n) = 2^(2p - 1)/2, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

16, 4096, 1152921504606846976, 7237005577332262213973186563042994240829374041602535252466099000494570602496

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

a(5) has 4931 digits and is too large to include. - R. J. Mathar, May 30 2008

Cf. A000668, A139286, A139294, A139295, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 4); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

a(n) = 2^(2*A000668(n)-1)/2.

One more term from R. J. Mathar, May 30 2008

A139297 a(n) = 2^(2p - 1)/2-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

15, 4095, 1152921504606846975, 7237005577332262213973186563042994240829374041602535252466099000494570602495

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A000668, A139294, A139295, A139296, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 4) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

a(n) = 2^(2*A000668(n)-1)/2-1 = A139296(n)-1.

a(4) from Amiram Eldar, Jul 10 2025

cp's OEIS Frontend

A248934 Decimal expansion of 2^3217 - 1, the 18th Mersenne prime A000668(18).

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A248935 Decimal expansion of 2^4253 - 1, the 19th Mersenne prime A000668(19).

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A248936 Decimal expansion of 2^4423 - 1, the 20th Mersenne prime A000668(20).

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A133049 Squares of Mersenne primes A000668(n).

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A138837 Non-Mersenne primes: A000040 \ A000668.

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A139295 a(n) = 2^(2p - 1)-1, where p is the n-th Mersenne prime A000668(n).

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A139231 First differences of Mersenne primes A000668.

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