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 1-7 of 7 results.

A138829 Bisection of imperfect numbers A132999.

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[(-x^14 + x^13 - x^4 + x^3 + x + 1)/(x^2 - 2*x + 1), {x, 0, 50}], x] (* G. C. Greubel, Feb 21 2017 *)
  • PARI
    x='x+O('x^50); Vec((-x^14 + x^13 - x^4 + x^3 + x + 1)/(x^2 - 2*x + 1)) \\ G. C. Greubel, Feb 21 2017

Formula

G.f.: (-x^14 + x^13 - x^4 + x^3 + x + 1)/(x^2 - 2*x + 1). - Alexander R. Povolotsky, Apr 06 2008

A138834 Bisection of even superperfect numbers A061652.

Original entry on oeis.org

2, 16, 4096, 262144, 1152921504606846976, 81129638414606681695789005144064
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

Also, bisection of superperfect numbers A019279, if there are no odd superperfect numbers.

Crossrefs

Formula

a(n) = A061652(2*n-1). - Jinyuan Wang, Mar 14 2020

A139567 Bisection of ultraperfect numbers A139306.

Original entry on oeis.org

8, 512, 33554432, 137438953472, 2658455991569831745807614120560689152
Offset: 1

Views

Author

Omar E. Pol, May 08 2008

Keywords

Crossrefs

Formula

a(n) = A139306(2*n-1). - Jinyuan Wang, Mar 14 2020

A139568 Bisection of ultraperfect numbers A139306.

Original entry on oeis.org

32, 8192, 8589934592, 2305843009213693952, 191561942608236107294793378393788647952342390272950272
Offset: 1

Views

Author

Omar E. Pol, May 08 2008

Keywords

Crossrefs

Formula

a(n) = A139306(2*n). - Jinyuan Wang, Mar 14 2020

A138830 Bisection of imperfect numbers A132999.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

The conjectured g.f. (-x^14+x^13-x^3+x^2+2)/(x^2-2*x+1) that a correspondent suggested is wrong and fails after roughly 240 terms [R. J. Mathar, Jun 15 2009]

Crossrefs

A138835 Bisection of even superperfect numbers A061652.

Original entry on oeis.org

4, 64, 65536, 1073741824, 309485009821345068724781056, 85070591730234615865843651857942052864
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

Also, bisection of superperfect numbers A019279, if there are no odd superperfect numbers.
Next terms have 183, 663 and more digits and are therefore not listed. - R. J. Mathar, May 22 2008

Crossrefs

A189373 Perfect numbers k such that k+1 is prime.

Original entry on oeis.org

6, 28, 33550336, 137438691328
Offset: 1

Views

Author

Luis H. Gallardo, Apr 23 2011

Keywords

Comments

Joerg Arndt checked that up to exponent p=110503 of the corresponding Mersenne prime 2^p - 1 the number k=2^(p-1)*(2^p-1)+1 is not pseudoprime.
The listed perfect numbers have exponents p in 2, 3, 13, 19.

Examples

			We have a(3) = 33550336 since 33550337 is prime and there is no other such perfect number less than a(3) and that exceeds a(2) = 28.
		

Crossrefs

Programs

  • PARI
    {e=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787];} /* exponents of Mersenne primes */
    for(n=1,#e,p=(2^e[n]-1)*(2^(e[n]-1));if(ispseudoprime(p+1),print1(p,", ")));

Formula

a(n) = A061644(n) - 1. - Amiram Eldar, May 06 2024
Showing 1-7 of 7 results.