cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.
Cf. A000043, A000668, A072868.

Programs

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

Formula

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

A139116 a(n) = p*(p-1)/2, where p is A000043(n).

Original entry on oeis.org

1, 3, 10, 21, 78, 136, 171, 465, 1830, 3916, 5671, 8001, 135460, 183921, 817281, 2425503, 2600340, 5172936, 9041878, 9779253, 46933516, 49406770, 62860078, 198732016, 235455850, 269317236, 989969256, 3718884403, 6105401253, 8718403176, 23347552095, 286402257541
Offset: 1

Views

Author

Omar E. Pol, May 10 2008

Keywords

Links

Crossrefs

Cf. A000043, A000217, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

Programs

  • Mathematica
    (#(#-1))/2&/@MersennePrimeExponent[Range[47]] (* Harvey P. Dale, Aug 13 2021 *)

Formula

a(n) = A000043(n)*(A000043(n)-1)/2.

Extensions

a(24)-a(32) from Harvey P. Dale, Aug 13 2021

A139115 a(n) = p*(p - 1), where p is A000043(n).

Original entry on oeis.org

2, 6, 20, 42, 156, 272, 342, 930, 3660, 7832, 11342, 16002, 270920, 367842, 1634562, 4851006, 5200680, 10345872, 18083756, 19558506, 93867032, 98813540, 125720156, 397464032, 470911700, 538634472, 1979938512, 7437768806, 12210802506
Offset: 1

Views

Author

Omar E. Pol, May 10 2008

Keywords

Links

Crossrefs

Cf. A000043, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

Programs

  • Mathematica
    #(#-1)&/@MersennePrimeExponent[Range[30]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2020 *)

Formula

a(n) = A000043(n)*(A000043(n) - 1).

Extensions

More terms from Vincenzo Librandi, May 11 2010
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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