Nathan Mabey - cp's OEIS frontend

User: Nathan Mabey

Nathan Mabey has authored 2 sequences.

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

Nathan Mabey, May 08 2022

nonn

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.

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

Nathan Mabey, C Script

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

C
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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

More terms from Jinyuan Wang, Jun 15 2022

A334261 Numbers m that are solutions of the arithmetic differential equation 2m" - m' - 4 = 0.

Original entry on oeis.org

4, 9, 21, 25, 33, 49, 57, 69, 85, 93, 121, 129, 133, 145, 169, 177, 205, 213, 217, 237, 249, 253, 265, 289, 309, 361, 393, 417, 445, 469, 489, 493, 505, 517, 529, 553, 565, 573, 597, 633, 669, 685, 697, 753, 781, 793, 813, 817, 841, 865, 889, 913, 933, 949, 961, 973, 985, 993

Offset: 1

Nathan Mabey, Apr 20 2020

Contains all the squared primes, plus additional terms.

At least some of these additional terms that are not divisible by 3 are given by the product of the smallest prime in a prime triplet, and the largest prime in that triplet, this certainly being true until the term 11413.

			4 is a term since 4'' = 4, 4' = 4, and 2*4 - 4 - 4 = 0.

Nathan Mabey, Table of n, a(n) for n = 1..20263

Cf. A003415 (n'), A068346 (n"), A001248 (squared primes), A368701 (characteristic function).

Cf. also A189941.

d(n) = if (n>0, my(f=factor(n)); sum(i=1, #f~, n*f[i, 2]/f[i, 1]), 0);
isok(m) = -2*d(d(m)) + d(m) + 4 == 0; \\ Michel Marcus, Apr 21 2020

from sympy import factorint
def d(n): return sum(n*e//p for p, e in factorint(n).items())
def ok(m): return m > 1 and 2*d(d(m)) - d(m) == 4
print([k for k in range(1, 999) if ok(k)]) # Michael S. Branicky, Aug 15 2022

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User: Nathan Mabey

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).

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