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 1-4 of 4 results.

A276493 Perfect numbers whose sum of prime factors is prime.

Original entry on oeis.org

6, 28, 8128, 14474011154664524427946373126085988481573677491474835889066354349131199152128
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 05 2016

Keywords

Comments

The next term is too large to include.
Numbers (2^n - 1)*2^(n - 1) such that both 2^n - 1 and 2^n + 2*n - 3 are prime.
Conjectures (defining x = 170141183460469231731687303715884105727 = A007013(4)):
(1) (2^x - 1)*2^(x - 1) is a term because 2^x - 1 and 2^x + 2*x - 3 are primes;
(2) a(n) is equal to (2^A007013(k) - 1)*2^(A007013(k) - 1) such that 2^A007013(k) - 1 and 2^A007013(k) + 2*A007013(k) - 3 are primes for some prime value of A007013(k) where k => 0;
(3) primes of A007013 are Mersenne prime exponents A000043, i.e. x is new exponent in A000043.

Examples

			a(1) = (2^2-1)*2^(2-1) = 6 because both 2^2-1 = 3 and 2^2+2*2-3 = 5 are primes.
a(2) = (2^3-1)*2^(3-1) = 28 because both 2^3-1 = 7 and 2^3+2*3-3 = 11 are primes.
a(3) = (2^7-1)*2^(7-1) = 8128 because both 2^7-1 = 127 and 2^7+2*7-3 = 139 are primes.

Crossrefs

Subsequence of A000396. Subsequence of A100118.
Cf. A000043, A000668, A007013, A192436, A276511.

Programs

  • Magma
    [(2^p-1)*2^(p-1): p in PrimesUpTo(2000) | IsPrime(2^p+2*p-3)];
  • Magma
    [(2^n-1)*2^(n-1): n in [1..200] | IsPrime(n) and IsPrime(2^n-1) and IsPrime(2^n+2*n-3)]; // Vincenzo Librandi, Sep 06 2016
  • Maple
    A276493:=n->`if`(isprime(n) and isprime(2^n-1) and isprime(2^n+2*n-3), (2^n-1)*2^(n-1), NULL): seq(A276493(n), n=1..10^3); # Wesley Ivan Hurt, Sep 07 2016
  • Mathematica
    Select[PerfectNumber[Range[12]],PrimeQ[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[#]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 06 2020 *)

A276511 Primes that are equal to the sum of the prime factors of some perfect number.

Original entry on oeis.org

5, 11, 139, 170141183460469231731687303715884105979
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 06 2016

Keywords

Comments

Primes of the form 2^n + 2*n - 3 such that 2^n - 1 is also prime.
Conjectures (defining x = 170141183460469231731687303715884105727 = A007013(4)):
(1) 2^x + 2*x - 3 is in this sequence;
(2) a(5) = 2^x + 2*x - 3 (see comments of A276493);
(3) primes of A007013 are Mersenne prime exponents A000043, i.e., x is new exponent in A000043.

Examples

			a(1) = 5 because 2^2-1 = 3 and 2^2+2*2-3 = 5 are primes,
a(2) = 11 because 2^3-1 = 7 and 2^3+2*3-3 = 11 are primes,
a(3) = 139 because 2^7-1 = 127 and 2^7+2*7-3 = 139 are primes.

Crossrefs

Subsequence of A192436.
Cf. A000043, A000396, A000668, A007013, A100118, A276493.

Programs

  • Magma
    [2^n+2*n-3: n in [1..200] | IsPrime(2^n-1) and IsPrime(2^n+2*n-3)];
  • Maple
    A276511:=n->`if`(isprime(2^n-1) and isprime(2^n+2*n-3), 2^n+2*n-3, NULL): seq(A276511(n), n=1..10^3); # Wesley Ivan Hurt, Sep 07 2016

Extensions

Name suggested by Michel Marcus, Sep 07 2016

A192764 Numbers k such that 2^(k-1)+2*k-1 is a prime number.

Original entry on oeis.org

1, 2, 6, 14, 66, 86, 230, 2006, 3876, 3920, 5418, 8820, 11900, 16669, 19446, 28243, 33408, 37919, 40595
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 09 2011

Keywords

Comments

a(20) > 10^5 if it exists. - Michael S. Branicky, Aug 26 2024

Crossrefs

Cf. A004051, A163115, A176691, A192436.

Programs

  • Mathematica
    Select[Range[4000], PrimeQ[2^(# - 1) + 2# - 1] &] (* Alonso del Arte, Jul 09 2011 *)
  • PARI
    for(n=1,10^6,if(ispseudoprime(2^(n-1)+2*n-1),print1(n,", ")));

Extensions

a(13)-a(19) from Michael S. Branicky, Jul 14 2023

A276663 Sum of primes dividing n-th perfect number (with repetition).

Original entry on oeis.org

5, 11, 39, 139, 8215, 131103, 524323, 2147483707, 2305843009213694071, 618970019642690137449562287, 162259276829213363391578010288339, 170141183460469231731687303715884105979
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 12 2016

Keywords

Comments

Numbers that are equal to the sum of the prime factors (A001414) of some perfect number.
The next term is too large to include.
A001222(a(n)) is 1, 1, 2, 1, 3, 4, 2, 2, 4, 6, 7, 1, 11, ...

Examples

			39 is in this sequence because 39 - 2^(5 - 1) = 31 = 2^5 - 1 and 31 is prime.

Links

Crossrefs

Subsequence of A131898. Supersequence of A276511.
Cf. A000043, A000396, A001222, A001414, A007013, A192436, A239546, A276493.

Programs

  • Mathematica
    Table[Total[Times@@@FactorInteger[PerfectNumber[n]]],{n,15}] (* Harvey P. Dale, Sep 22 2019 *)
  • PARI
    \\ Ochem & Rao: no odd perfect numbers below 10^1500
forprime(p=2,2281, if(ispseudoprime(t=2^p-1), print1(2^p+2*p-3", "))) \\ Charles R Greathouse IV, Sep 18 2016

Formula

a(n) = 2^A000043(n) + 2*A000043(n) - 3, assuming that there are no odd perfect numbers.
a(n) = A001414(A000396(n)). - Michel Marcus, Sep 18 2016
Showing 1-4 of 4 results.

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