A000668 -id:A000668 - cp's OEIS frontend

A330820 Numbers of the form (M_p^2-1)^2, where M_p is a Mersenne prime, A000668. Also the second element of the power-spectral basis of A330817.

64, 2304, 921600, 260112384, 4501400872550400, 295138898048817561600, 75557287266261531623424

Offset: 1

Walter Kehowski, Jan 06 2020

nonn

The first element of the power-spectral basis of A330817 is A330819.

			If n=1, a(1)=(3^2-1)^2=64.

Cf. A000043, A000668, A330817, A330818, A330819.

A330820:=[]:
for www to 1 do
for i from 1 to 31 do
#ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
if isprime(q) then x:=(q^2-1)^2; A330820:=[op(A330820),x] fi;
od;
od;
A330820;

Array[((2^MersennePrimeExponent[#] - 1)^2 - 1)^2 &, 10] (* Amiram Eldar, Jan 07 2020 *)

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

A139257 Twice Mersenne primes A000668(n).

Original entry on oeis.org

6, 14, 62, 254, 16382, 262142, 1048574, 4294967294, 4611686018427387902, 1237940039285380274899124222, 324518553658426726783156020576254, 340282366920938463463374607431768211454

Offset: 1

Omar E. Pol, Apr 23 2008

nonn

Radicals of even perfect numbers. - Charles R Greathouse IV, Feb 01 2013

Amiram Eldar, Table of n, a(n) for n = 1..18
Florian Luca and Carl Pomerance, On the radical of a perfect number, New York Journal of Math., 16 (2010), 23-30; alternative link.

Cf. A000043, A000668, A000918, A060590, A095121, A100484, A139256.

2*(2^MersennePrimeExponent[Range[15]]-1) (* Harvey P. Dale, Jan 05 2020 *)

apply(p->2*(2^p-1),select(p->ispseudoprime(2^p-1),primes(40))) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = 2*A000668(n).

a(n) = A000918(1 + A000043(n)) = A095121(A000043(n)). - Omar E. Pol, Jun 07 2012

Corrected and extended by Joerg Arndt, Jun 07 2012.

A275977 Decimal expansion of 2^9689 - 1, the 21st Mersenne prime A000668(21).

Original entry on oeis.org

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

Offset: 2917

Arkadiusz Wesolowski, Aug 15 2016

			47822027880546120295283929866000590974149717240223650085133451099183789...

Arkadiusz Wesolowski, Table of n, a(n) for n = 2917..5833
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275982 = A000668(25), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^9689-1))[1..105];
```

First@RealDigits@N[2^9689 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)
RealDigits[2^MersennePrimeExponent[21]-1,10,120][[1]] (* Harvey P. Dale, Aug 14 2025 *)

PARI
```
eval(Vec(Str(2^9689-1)))[1..105]
```

2^A000043(21) - 1.

A275979 Decimal expansion of 2^9941 - 1, the 22nd Mersenne prime A000668(22).

Original entry on oeis.org

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

Offset: 2993

Arkadiusz Wesolowski, Aug 15 2016

			34608828249085121524296039576741331672262866890023854779048928344500622...

Arkadiusz Wesolowski, Table of n, a(n) for n = 2993..5985
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275980 = A000668(23), A275981 = A000668(24), A275982 = A000668(25), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^9941-1))[1..105];
```

First@RealDigits@N[2^9941 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^9941-1)))[1..105]
```

2^A000043(22) - 1.

A275980 Decimal expansion of 2^11213 - 1, the 23rd Mersenne prime A000668(23).

Original entry on oeis.org

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

Offset: 3376

Arkadiusz Wesolowski, Aug 15 2016

			28141120136973731333931529758425841918186623820136007878924193493455151...

Arkadiusz Wesolowski, Table of n, a(n) for n = 3376..6751
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275981 = A000668(24), A275982 = A000668(25), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^11213-1))[1..105];
```

First@RealDigits@N[2^11213 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^11213-1)))[1..105]
```

2^A000043(23) - 1.

A275981 Decimal expansion of 2^19937 - 1, the 24th Mersenne prime A000668(24).

Original entry on oeis.org

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

Offset: 6002

Arkadiusz Wesolowski, Aug 15 2016

			43154247973881626480552355163379198390539350432267115051652505414033306...

Arkadiusz Wesolowski, Table of n, a(n) for n = 6002..12003
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275982 = A000668(25), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^19937-1))[1..105];
```

First@RealDigits@N[2^19937 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^19937-1)))[1..105]
```

2^A000043(24) - 1.

A275982 Decimal expansion of 2^21701 - 1, the 25th Mersenne prime A000668(25).

Original entry on oeis.org

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

Offset: 6533

Arkadiusz Wesolowski, Aug 15 2016

			44867916611904333479495141036159177872720902372938861301036480447512785...

Arkadiusz Wesolowski, Table of n, a(n) for n = 6533..13065
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^21701-1))[1..105];
```

First@RealDigits@N[2^21701 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^21701-1)))[1..105]
```

2^A000043(25) - 1.

A275983 Decimal expansion of 2^23209 - 1, the 26th Mersenne prime A000668(26).

Original entry on oeis.org

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

Offset: 6987

Arkadiusz Wesolowski, Aug 15 2016

			40287411577898877818187332907159176772243850689162242004102996357869459...

Arkadiusz Wesolowski, Table of n, a(n) for n = 6987..13973
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275982 = A000668(25), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^23209-1))[1..105];
```

First@RealDigits@N[2^23209 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^23209-1)))[1..105]
```

2^A000043(26) - 1.

A275984 Decimal expansion of 2^44497 - 1, the 27th Mersenne prime A000668(27).

Original entry on oeis.org

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

Offset: 13395

Arkadiusz Wesolowski, Aug 15 2016

			85450982430363380319330070531840303650990159130402105834326925828229006...

Arkadiusz Wesolowski, Table of n, a(n) for n = 13395..26789
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275982 = A000668(25), A275983 = A000668(26).

Magma
```
Reverse(Intseq(2^44497-1))[1..105];
```

First@RealDigits@N[2^44497 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^44497-1)))[1..105]
```

2^A000043(27) - 1.

A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

Original entry on oeis.org

326, 748, 166, 748, 166, 166, 748, 748, 166, 166, 748, 748, 166, 748, 748, 748, 166, 166, 166, 748, 166, 166, 166, 166, 166, 166, 166, 748, 748, 166, 748, 748, 166, 748, 166, 166, 166, 166, 166

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th superperfect number A019279(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of n-th term of A080172, A138125(n) and A094540(n).
a(1)=326. For n>1 a(n) is equal to 166 or 748, only.

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138817.

cp's OEIS Frontend

A330820 Numbers of the form (M_p^2-1)^2, where M_p is a Mersenne prime, A000668. Also the second element of the power-spectral basis of A330817.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A139257 Twice Mersenne primes A000668(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A275977 Decimal expansion of 2^9689 - 1, the 21st Mersenne prime A000668(21).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A275979 Decimal expansion of 2^9941 - 1, the 22nd Mersenne prime A000668(22).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A275980 Decimal expansion of 2^11213 - 1, the 23rd Mersenne prime A000668(23).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A275981 Decimal expansion of 2^19937 - 1, the 24th Mersenne prime A000668(24).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A275982 Decimal expansion of 2^21701 - 1, the 25th Mersenne prime A000668(25).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs