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

14, 2, 1, 3, 6, 9, 5, 7, 62

Offset

0,1

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

Links

Table of n, a(n) for n=0..8.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000203, A007369, A112645, A112646.

Sequence in context: A350888 A348335 A040192 * A040191 A040193 A064972

Adjacent sequences: A112644 A112645 A112646 * A112648 A112649 A112650

Keyword

nonn,more

Author

Labos Elemer, Sep 28 2005

Extensions

Entry revised by N. J. A. Sloane, May 25 2016

a(0) prepended by Giovanni Resta, Oct 29 2019

Status

approved