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

1,1

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.

Links

Table of n, a(n) for n=1..47.

Nathan Mabey, C Script

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A045554 A174649 A161452 * A212867 A178494 A049725

Adjacent sequences: A353869 A353870 A353871 * A353873 A353874 A353875

Keyword

nonn

Author

Nathan Mabey, May 08 2022

Extensions

More terms from Jinyuan Wang, Jun 15 2022

Status

approved