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A370126 Numbers k for which k’ = x’*y’, where k = x + y with x and y composite, and k’, x’, y’ are the arithmetic derivatives of k, x, y.

%I A370126 #10 Feb 22 2024 20:09:06
%S A370126 48,160,208,212,236,252,268,496,612,752,1040,1172,1376,1476,1568,1584,
%T A370126 1692,1728,2016,2192,2736,3616,3627,3632,3760,3932,4048,4508,4572,
%U A370126 5264,5392,5900,6224,6336,6640,6652,6948,7376,7648,8244,8928,9072,9392,9712,10648,10736,10960,12500,12544,12592,12960,13284,16452
%N A370126 Numbers k for which k’ = x’*y’, where k = x + y with x and y composite, and k’, x’, y’ are the arithmetic derivatives of k, x, y.
%H A370126 Antti Karttunen, <a href="/A370126/b370126.txt">Table of n, a(n) for n = 1..269</a>
%e A370126 48 is included as 48 = 15+33, and 15' * 33' = 8*14 = 112 = 48' = A003415(48).
%e A370126 1728 (= 2^6 * 3^3) is included as 1728 = 4+1724, and 4' * 1724' = 4*1728 = 6912 = 1728'.
%e A370126 3627 is included as 3627 = 38+3589, and 38' * 3589' = 21*134 = 2814 = 3627'.
%o A370126 (PARI)
%o A370126 up_to = 2^18;
%o A370126 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A370126 v003415 = vector(up_to,n,A003415(n));
%o A370126 isA370126(n) = { my(z=v003415[n]); for(x=2,ceil(n/2),if(!isprime(x) && !isprime(n-x) && !(z%v003415[x]), if(z==v003415[x]*v003415[n-x], return(1)))); (0); }; \\ _Antti Karttunen_, Feb 22 2024
%Y A370126 Cf. A003415.
%Y A370126 Subsequence of A218011.
%K A370126 nonn
%O A370126 1,1
%A A370126 _Antti Karttunen_, Feb 21 2024