A203617 Numbers m such that (m'-1)' = m+1, where m' denotes the arithmetic derivative of m.

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

Offset: 1

Author

Paolo P. Lava, Jan 20 2012

Keywords

• nonn
• more

Comments

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.

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.

A007850 is a subsequence of this sequence.

a(11) > 10^9. - Michel Marcus, Nov 05 2014

a(12) > 10^10. - Giovanni Resta, Jun 04 2016

Examples

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

Crossrefs

Cf. A007850, A185222, A203618.

Programs

Maple

with(numtheory);
P:=proc(i)
local a,n,p,pfs;
for n from 1 to i do
  pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs) ;
  pfs:=ifactors(a-1)[2]; a:=(a-1)*add(op(2,p)/op(1,p),p=pfs) ;
  if a=n+1 then print(n); fi;
od;
end:
P(10000000);

PARI

ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
isok(n) = my(m = ad(n)-1); (m) && ad(m) == n+1; \\ Michel Marcus, Nov 05 2014

Extensions

a(11) from Giovanni Resta, Jun 04 2016