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.

A028491 Numbers k such that (3^k - 1)/2 is prime.

Original entry on oeis.org

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117
Offset: 1

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Author

N. J. A. Sloane, Jean-Yves Perrier (nperrj(AT)ascom.ch)

Keywords

Comments

If k is in the sequence and m=3^(k-1) then m is a term of A033632 (phi(sigma(m)) = sigma(phi(m))), so 3^(A028491-1) is a subsequence of A033632. For example since 9551 is in the sequence, phi(sigma(3^9550)) = sigma(phi(3^9550)). - Farideh Firoozbakht, Feb 09 2005
Salas lists these, except 3, in "Open Problems" p. 6 [March 2012], and proves that the Cantor primes > 3 are exactly the prime-valued cyclotomic polynomials of the form Phi_s(3^{s^j}) == 1 (mod 4).
Also, k such that 3^k-1 is a semiprime - see also A080892. - M. F. Hasler, Mar 19 2013

References

  • J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A076481, A033632, A112646.

Programs

Extensions

a(13) from Farideh Firoozbakht, Mar 27 2005
a(14)-a(16) from Robert G. Wilson v, Apr 11 2005
All larger terms only correspond to probable primes.
a(17) from Paul Bourdelais, Feb 08 2010
a(18) from Paul Bourdelais, Jul 06 2010
a(19) from Paul Bourdelais, Feb 05 2019
a(20) and a(21) from Ryan Propper, Dec 29 2021
a(22) from Ryan Propper, Nov 06 2023
a(23) from Ryan Propper, Nov 09 2023

A112645 Solutions to abs(sigma(x+1) - sigma(x)) = 2. Divisor sums of x and its neighbor x+1 differ from each other by 2.

Original entry on oeis.org

1, 8, 26, 117, 2186, 145215, 1594322
Offset: 1

Views

Author

Labos Elemer, Sep 28 2005

Keywords

Comments

Observe that form of 8, 26, 2186 and 1594322 is -1+3^j. Exponents of powers of 3 suitable as number n+1 are as follows: 2, 3, 7, 13, 71, 103. Is the next term 7509466514979724803946715958257546 = -1 + 3^71?
a(8) > 10^13. - Giovanni Resta, Jul 11 2013

Examples

			n = 1594322 = 2*797161 while n+1 = 3^13;
Sigma(n) = 2391486, sigma(n+1) = 2391484.

Crossrefs

Cf. A000203, A112646, A112647.

Programs

  • Mathematica
    ta={{0}};Do[s=Abs[DivisorSigma[1, n+1]-DivisorSigma[1, n]]; If[Equal[s, 2], ta=Append[ta, n];Print[n]], {n, 1, 100000000}]; ta=Delete[ta, 1]

A112647 a(n)=x is the smallest solution to abs(sigma(x+1)-sigma(x))=n, or 0 if no solution exists.

Original entry on oeis.org

14, 2, 1, 3, 6, 9, 5, 7, 62
Offset: 0

Views

Author

Labos Elemer, Sep 28 2005

Keywords

Comments

The known values plus conjectured zero values are: 14, 2, 1, 3, 6, 9, 5, 7, 62, 0, 13, 25, 22, 16, 12, 32, 11, 0, 104, 18, 837, 17, 19, 63, 46, 0, 28, 0, 116, 24, 58, 31, 2222, 0, 39, 242, 23, 0, 147, 0, 30, 675, 29, 35, 52, 0, 777, 0, 40, 0, 435, 0, 42, 36, 41, 0, 91, 0, 67, 0, 65, 99, 0, 195, 110, 80, 53, 48, 124, 0, 243, 0, 70, 97.
The first unknown value is a(9).
The zero values are based on a search up to 10000000.
While it is known that not all m values satisfy sigma(x) = m (see A007369), it is more difficult to determine those numbers which cannot be a difference of sigma(u)-sigma(w) for some u and w.
No solutions to abs(sigma(x+1)-sigma(x)) = n with x < 2*10^8 for n = 9, 17, 25, 27, 33, 37, 39, 45, 47, 49, 51, 55, 57, 59, 62, 69, 71. - Robert Israel, May 24 2016
Except for a(62) = 1159742042, all the terms a(n)>0 with n <= 100 are either smaller than 10^6 or greater than 2*10^12. - Giovanni Resta, Oct 29 2019

Examples

			n=5: least solution is 9 because sigma for 9 and 9+1=10 are 13 and 13+5=18.

Crossrefs

Cf. A000203, A007369, A112645, A112646.

Programs

  • MATLAB
    N = 2*10^8; % to search sigma(n) for n <= N
M = 100;    % to get a(1) to a(M)
Sigma = ones(1,N);
for n=2:N
  inds = [n:n:N];
  Sigma(inds) = Sigma(inds) + n;
end
DSigma = abs(Sigma(2:end) - Sigma(1:end-1));
A = zeros(1,M);
for v = 1:M
  r = find(DSigma == v,1,'first');
  if numel(r) > 0
    A(v) = r;
  end
end
A   % Robert Israel, May 24 2016
  • Mathematica
    f[x_] :=Abs[DivisorSigma[1, n+1] - DivisorSigma[1, n]]; t=Table[0, {258}]; Do[s=f[n]; If[s<258 && t[[s+1]]==0, t[[s+1]]=n], {n, 10^7}]; t (* edited by Giovanni Resta, Oct 29 2019 *)

Extensions

Entry revised by N. J. A. Sloane, May 25 2016
a(0) prepended by Giovanni Resta, Oct 29 2019
Showing 1-3 of 3 results.

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

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