A203617 - OEIS

%I #29 Dec 17 2024 18:30:37

%S 30,210,246,858,1722,66198,235290,282342,1929378,1976394,2214408306

%N Numbers m such that (m'-1)' = m+1, where m' denotes the arithmetic derivative of m.

%C The differential equation whose solutions are the Giuga numbers is m' = k*m+1, with k a positive integer. Let us rewrite the equation as m'-1 = k*m and then take the derivative: (m'-1)' = (k*m)' = k'*m + k*m' = k'*m + k*(k*m+1) = (k'+k^2)*m+k.

%C Let k=1: (m'-1)' = m+1. The solutions of this equation are the Giuga numbers plus pairs of numbers (x,y) for which x' = y+1 and y' = x+1.

%C A007850 is a subsequence of this sequence.

%C a(11) > 10^9. - _Michel Marcus_, Nov 05 2014

%C a(12) > 10^10. - _Giovanni Resta_, Jun 04 2016

%e 235290' = 282343; (282343 - 1)' = 282342' = 235291 = 235290 + 1, so 235290 is a term.

%e 282342' = 235291; (235291 - 1)' = 235290' = 282343 = 282342 + 1, so 282342 is a term.

%p with(numtheory);

%p P:=proc(i)

%p local a,n,p,pfs;

%p for n from 1 to i do

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

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

%p   if a=n+1 then print(n); fi;

%p od;

%p end:

%p P(10000000);

%o (PARI) ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);

%o isok(n) = my(m = ad(n)-1); (m) && ad(m) == n+1; \\ _Michel Marcus_, Nov 05 2014

%Y Cf. A007850, A185222, A203618.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Jan 20 2012

%E a(11) from _Giovanni Resta_, Jun 04 2016