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

Author

Labos Elemer, Sep 28 2005

Keywords

• nonn
• more
• hard

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