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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).
1
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.
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
KEYWORD
nonn
AUTHOR
Nathan Mabey, May 08 2022
EXTENSIONS
More terms from Jinyuan Wang, Jun 15 2022
STATUS
approved