A212662 - OEIS

%I #14 Sep 02 2022 08:02:48

%S 3,6,9,12,15,18,21,24,25,27,30,33,36,39,42,45,48,50,51,54,55,57,60,63,

%T 66,69,72,75,78,81,82,84,85,87,90,93,95,96,99,100,102,105,108,110,111,

%U 114,116,117,120,121,123,125,126,129,132,135,138,141,144,145

%N Numbers k for which k' = x' + y', where x > 0, k = x + y, and k', x', y' are the arithmetic derivatives of k, x, y.

%H Paolo P. Lava, <a href="/A212662/b212662.txt">Table of n, a(n) for n = 1..5000</a>

%e k=24, x=8, y=16 and 24=8+16; k'=44, x'=12, y'=32 and 44=12+32.

%e In more than one way:

%e k=39, x=4, y=35 and 39=4+35; k'=16, x'=4, y'=12 and 16=4+12;

%e k=39, x=13, y=26 and 39=13+26; k'=16, x’=1, y'=15 and 16=1+15.

%e k=255, x=54, y=201 and 255=54+201; k'=151, x'=81, y'=70 and 16=4+12;

%e k=255, x=85, y=170 and 255=85+170; k'=151, x'=22, y'=129 and 16=1+15;

%e k=255, x=114, y=141 and 39=13+26; k'=151, x'=101, y'=50 and 16=1+15.

%p with(numtheory);

%p A212662:=proc(q)

%p local a,b,c,i,n,p,pfs;

%p for n from 1 to q do

%p   pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs);

%p   for i from 1 to trunc(n/2) do

%p     pfs:=ifactors(i)[2]; b:=i*add(op(2,p)/op(1,p),p=pfs);

%p     pfs:=ifactors(n-i)[2]; c:=(n-i)*add(op(2,p)/op(1,p),p=pfs);

%p     if a=b+c then print(n); break; fi;

%p   od;

%p od; end:

%p A212662(1000);

%o (PARI) ard(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415

%o isok(m) = for (k=1, m\2, if (ard(m-k)+ard(k) == ard(m), return(1))); \\ _Michel Marcus_, Aug 27 2022

%Y Cf. A003415, A212663, A212664.

%Y Cf. A211223, A211224, A211225.

%K nonn

%O 1,1

%A _Paolo P. Lava_, May 23 2012