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
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
KEYWORD
nonn,more,hard
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