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

1,1

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

Links

Table of n, a(n) for n=1..11.

Example

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.

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);

Prog

(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

Crossrefs

Cf. A007850, A185222, A203618.

Sequence in context: A209789 A280482 A371721 * A181629 A346245 A129499

Adjacent sequences: A203614 A203615 A203616 * A203618 A203619 A203620

Keyword

nonn,more

Author

Paolo P. Lava, Jan 20 2012

Extensions

a(11) from Giovanni Resta, Jun 04 2016

Status

approved