A353872 Numbers k such that the arithmetic differential equation m'' - m'm + k = 0 has exactly one positive solution in m with two prime factors (counted with multiplicity).

12, 29, 49, 69, 108, 120, 203, 243, 285, 382, 404, 453, 592, 645, 677, 788, 848, 996, 1140, 1149, 1241, 1365, 1779, 1796, 1797, 1857, 2032, 2236, 2649, 2704, 2812, 2870, 3143, 3188, 3388, 3443, 3525, 3831, 4372, 4379, 4592, 4799, 4911, 5204, 5364, 5520, 5814

Offset: 1

Author

Nathan Mabey, May 08 2022

Keywords

• nonn

Comments

This is a second-order nonlinear ADE. It is known that many linear second-order ADEs have infinitely many solutions (A334261), but nonlinear cases haven't been studied.

Examples

			k = 12 is in the sequence, since for m = 4, we have m' = m'' = 4, so m'm - m'' = 16 - 4 = 12 = k.
		

Links

• Nathan Mabey, C Script

Crossrefs

Cf. A003415 (n'), A068346 (n''), A334261 (2m'' - m' - 4 = 0).

Programs

C

See Link

MATLAB

function a = A353872( max_pow_2 )
    a = [];
    maxad2 = ad(ad(2^max_pow_2));
    for m = 1:2^max_pow_2
        if length(factor(m)) == 2
            d = ad(m); b = ad(d); c = d*m;
            k(m) = b - c;
        end
    end
    for n = 1:length(k)
        if k(n) > -maxad2;
            if isempty(find(a == k(n),1))
                if 1 == length(find(k == k(n)))
                   a = [a k(n)];
                end
            end
        end
    end
    a = sort(-a);
end
function y = ad( x )
    y = 0;
    if(x > 1)
        p = factor(x); pu = unique(p);
        for n = 1:length(pu);
            y = y + (x*length(find(p == pu(n))))/pu(n);
        end
    end
end % Thomas Scheuerle, Jun 15 2022

Extensions

More terms from Jinyuan Wang, Jun 15 2022