A139286 -id:A139286 - cp's OEIS frontend

A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Sum of n-th even perfect number and n-th even superperfect number.

Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).

			a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.

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

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.

Cf. A000043, A000668, A072868.

2^(2 * MersennePrimeExponent[Range[10]] - 1) (* Amiram Eldar, Oct 17 2024 *)

a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).

a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - Omar E. Pol, Apr 13 2008

A139294 a(n) = 2^(2p - 1), where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

32, 8192, 2305843009213693952, 14474011154664524427946373126085988481658748083205070504932198000989141204992

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Next terms have 4932, 78913, 315652, 1292913986, and 1388255822130839283 decimal digits. - Jens Kruse Andersen, Jul 14 2014

Cf. A000079, A000668, A139286, A139295, A139296, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

A000668 := Select[2^Range[500] - 1, PrimeQ]; Table[2^(2*A000668[[n]] - 1), {n, 1, 5}] (* G. C. Greubel, Oct 03 2017 *)
a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 3); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

\p 100
print1("a(n): "); forprime(q=2, 7, p=2^q-1; if(isprime(p), print1(2^(2*p-1)", ")));
print1("\nNumber of digits in a(n): "); forprime(q=2, 127, p=2^q-1; if(isprime(p), print1(ceil((2*p-1)*log(2)/log(10))", "))) \\ Jens Kruse Andersen, Jul 14 2014

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

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

A139305 a(n) = (2^(2p - 1)/32)-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

0, 255, 72057594037927935, 452312848583266388373324160190187140051835877600158453279131187530910662655

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Next term is too large to list here.

Cf. A000668, A139286, A139294, A139295, A139296, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139306.

a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 8) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

a(n) = (2^(2*A000668(n)-1)/32)-1 = (A139294(n)/32)-1.

Edited by Max Alekseyev, Apr 23 2010

A139287 2^(2p - 1) - 1, where p is prime.

Original entry on oeis.org

7, 31, 511, 8191, 2097151, 33554431, 8589934591, 137438953471, 35184372088831, 144115188075855871, 2305843009213693951, 9444732965739290427391, 2417851639229258349412351, 38685626227668133590597631

Offset: 1

Omar E. Pol, Apr 13 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040, A000079, A076274, A139286, A139288, A139289, A139290, A139291, A139292, A139293, A139294, A139306.

[2^(2*p-1)-1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010

Table[(2^(2 Prime[n] - 1) - 1), {n, 1, 20}] (* Vincenzo Librandi, May 24 2014 *)

a(9) corrected by Vincenzo Librandi, May 24 2014

A139290 2^(2p - 1)/4, where p is prime.

Original entry on oeis.org

2, 8, 128, 2048, 524288, 8388608, 2147483648, 34359738368, 8796093022208, 36028797018963968, 576460752303423488, 2361183241434822606848, 604462909807314587353088, 9671406556917033397649408

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A139286, A139287, A139288, A139289, A139290, A139292, A139293, A139294, A139306.

[2^(2*p-1)div 4: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010

2^(2#-1)/4 & /@ Prime@ Range@ 15 (* Harvey P. Dale, Dec 16 2010 *)

a(n) = A139286(n)/4.

More terms from Jon E. Schoenfield, Jul 16 2010

A139288 2^(2p - 1)/2, where p is prime.

Original entry on oeis.org

4, 16, 256, 4096, 1048576, 16777216, 4294967296, 68719476736, 17592186044416, 72057594037927936, 1152921504606846976, 4722366482869645213696, 1208925819614629174706176, 19342813113834066795298816

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A139286, A139287, A139289, A139290, A139291, A139292, A139293, A139294, A139306.

Table[2^(2p-1)/2,{p,Prime[Range[20]]}] (* Harvey P. Dale, Jun 21 2021 *)

a(n)=A139286(n)/2.

More terms from Jon E. Schoenfield, Jul 16 2010

A139289 (2^(2p - 1)/2)-1, where p is prime.

Original entry on oeis.org

3, 15, 255, 4095, 1048575, 16777215, 4294967295, 68719476735, 17592186044415, 72057594037927935, 1152921504606846975, 4722366482869645213695, 1208925819614629174706175, 19342813113834066795298815, 4951760157141521099596496895, 20282409603651670423947251286015, 83076749736557242056487941267521535

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Floris P. van Doorn and Jasper Mulder, Table of n, a(n) for n=1..260.

Cf. A139286, A139287, A139288, A139290, A139291, A139292, A139293, A139294, A139306.

Table[(2^(2 Prime[p] - 1)/2) - 1, {p, 1, 20}] (* Floris P. van Doorn and Jasper Mulder (florisvandoorn(AT)hotmail.com), Oct 12 2009 *)

a(n) = A139286(n)/2 - 1.

More terms from Floris P. van Doorn and Jasper Mulder (florisvandoorn(AT)hotmail.com), Oct 12 2009

A139292 2^(2p - 1)/8, where p is prime.

Original entry on oeis.org

1, 4, 64, 1024, 262144, 4194304, 1073741824, 17179869184, 4398046511104, 18014398509481984, 288230376151711744, 1180591620717411303424, 302231454903657293676544, 4835703278458516698824704

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A139286, A139287, A139288, A139289, A139290, A139291, A139293, A139294, A139306.

2^(2#-1)/8&/@Prime[Range[20]] (* Harvey P. Dale, Aug 19 2012 *)

a(n) = A139286(n)/8.

More terms from Jon E. Schoenfield, Jul 16 2010

A139293 (2^(2p - 1)/8)-1, where p is prime.

Original entry on oeis.org

0, 3, 63, 1023, 262143, 4194303, 1073741823, 17179869183, 4398046511103, 18014398509481983, 288230376151711743, 1180591620717411303423, 302231454903657293676543, 4835703278458516698824703

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A139286, A139287, A139288, A139289, A139290, A139291, A139292, A139294, A139306.

(2^(2#-1))/8-1&/@Prime[Range[30]] (* Harvey P. Dale, May 09 2012 *)

a(n) = (A139286(n)/8)-1.

More terms from Jon E. Schoenfield, Jul 16 2010

cp's OEIS Frontend

A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

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A139294 a(n) = 2^(2p - 1), where p is the n-th Mersenne prime A000668(n).

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

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A139305 a(n) = (2^(2p - 1)/32)-1, where p is the n-th Mersenne prime A000668(n).

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A139287 2^(2p - 1) - 1, where p is prime.

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A139290 2^(2p - 1)/4, where p is prime.

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A139288 2^(2p - 1)/2, where p is prime.

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A139289 (2^(2p - 1)/2)-1, where p is prime.

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