cp's OEIS Frontend

A212662 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.

%I A212662 #14 Sep 02 2022 08:02:48
%S A212662 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 A212662 66,69,72,75,78,81,82,84,85,87,90,93,95,96,99,100,102,105,108,110,111,
%U A212662 114,116,117,120,121,123,125,126,129,132,135,138,141,144,145
%N A212662 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 A212662 Paolo P. Lava, <a href="/A212662/b212662.txt">Table of n, a(n) for n = 1..5000</a>
%e A212662 k=24, x=8, y=16 and 24=8+16; k'=44, x'=12, y'=32 and 44=12+32.
%e A212662 In more than one way:
%e A212662 k=39, x=4, y=35 and 39=4+35; k'=16, x'=4, y'=12 and 16=4+12;
%e A212662 k=39, x=13, y=26 and 39=13+26; k'=16, x’=1, y'=15 and 16=1+15.
%e A212662 k=255, x=54, y=201 and 255=54+201; k'=151, x'=81, y'=70 and 16=4+12;
%e A212662 k=255, x=85, y=170 and 255=85+170; k'=151, x'=22, y'=129 and 16=1+15;
%e A212662 k=255, x=114, y=141 and 39=13+26; k'=151, x'=101, y'=50 and 16=1+15.
%p A212662 with(numtheory);
%p A212662 A212662:=proc(q)
%p A212662 local a,b,c,i,n,p,pfs;
%p A212662 for n from 1 to q do
%p A212662   pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs);
%p A212662   for i from 1 to trunc(n/2) do
%p A212662     pfs:=ifactors(i)[2]; b:=i*add(op(2,p)/op(1,p),p=pfs);
%p A212662     pfs:=ifactors(n-i)[2]; c:=(n-i)*add(op(2,p)/op(1,p),p=pfs);
%p A212662     if a=b+c then print(n); break; fi;
%p A212662   od;
%p A212662 od; end:
%p A212662 A212662(1000);
%o A212662 (PARI) ard(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A212662 isok(m) = for (k=1, m\2, if (ard(m-k)+ard(k) == ard(m), return(1))); \\ _Michel Marcus_, Aug 27 2022
%Y A212662 Cf. A003415, A212663, A212664.
%Y A212662 Cf. A211223, A211224, A211225.
%K A212662 nonn
%O A212662 1,1
%A A212662 _Paolo P. Lava_, May 23 2012