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A218011 Numbers n for which n’ = x’*y’, where x>0, y>0, n = x + y and n’, x’, y’ are the arithmetic derivatives of n, x, y.

%I A218011 #24 Feb 22 2024 20:08:06
%S A218011 5,7,13,19,31,43,48,55,61,73,74,87,103,106,109,117,139,146,151,159,
%T A218011 160,178,181,193,199,202,208,212,225,229,236,241,252,267,268,271,283,
%U A218011 285,298,313,349,357,362,386,403,411,421,433,455,463,496,511,519,523,535
%N A218011 Numbers n for which n’ = x’*y’, where x>0, y>0, n = x + y and n’, x’, y’ are the arithmetic derivatives of n, x, y.
%C A218011 The greatest prime in a twin primes couple is in the sequence. In fact if the twin primes are a and b, with a<b, b can be written as b=a+2. Being a’=b’=2’=1 we have b’=a’*2’ that is 1=1*1.
%H A218011 Antti Karttunen, <a href="/A218011/b218011.txt">Table of n, a(n) for n = 1..12822</a> (first 250 terms from Paolo P. Lava)
%e A218011 n= 612, x=85,  y=527; n’=1056, x’=22, y’=48 and 1056=22*48.
%e A218011 n= 752, x=361, y=391; n’=1520, x’=38, y’=40 and 1520=38*40.
%e A218011 n= 779, x=36,  y=743; n’=60,   x’=60, y’=1  and 60=60*1.
%p A218011 with(numtheory);
%p A218011 A218011:= proc(i)
%p A218011 local a,b,c,n,p,pfs,q;
%p A218011 for n from 1 to i do
%p A218011 for q from 1 to trunc(n/2) do
%p A218011   a:=q*add(op(2,p)/op(1,p),p= ifactors(q)[2]);
%p A218011   b:=(n-q)*add(op(2,p)/op(1,p),p= ifactors(n-q)[2]);
%p A218011   c:=n*add(op(2,p)/op(1,p),p= ifactors(n)[2]);
%p A218011   if c=a*b then lprint(n,q,n-q); break; fi;
%p A218011 od; od;
%p A218011 end:
%p A218011 A218011(1000000);
%t A218011 dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; f[n_] := Select[Range[n/2], dn[#]*dn[n - #] == dn[n] &]; Select[Range[535], Length[f[#]] > 0 &] (* _T. D. Noe_, Oct 18 2012 *)
%o A218011 (PARI)
%o A218011 up_to = 2^18;
%o A218011 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A218011 v003415 = vector(up_to,n,A003415(n));
%o A218011 isA218011(n) = { my(z=v003415[n]); for(x=2,ceil(n/2),if(!(z%v003415[x]), if(z==v003415[x]*v003415[n-x], return(1)))); (0); }; \\ _Antti Karttunen_, Feb 22 2024
%Y A218011 Cf. A003415, A211223, A211224, A211225, A212662, A212663, A212664.
%Y A218011 Subsequences: A006512 (primes in this sequence), A370126 (k with a solution where both x and y are composite).
%K A218011 nonn
%O A218011 1,1
%A A218011 _Paolo P. Lava_, Oct 18 2012