A203618 Numbers m such that (m'+1)' = m-1, where m' is the arithmetic derivative of m.

1, 2, 6, 42, 104, 120, 165, 245, 272, 561, 1806, 47058, 765625, 1137501, 3874128, 9131793, 2214502422, 52495396602

Offset: 1

Author

Paolo P. Lava, Jan 20 2012

Keywords

• nonn
• more

Comments

The differential equation whose solutions are the primary pseudoperfect 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 primary pseudoperfect numbers plus pairs of numbers (x,y) for which x' = y-1 and y' = x-1.

A054377 is a subsequence of this sequence.

a(17) > 10^9. - Michel Marcus, Nov 05 2014

a(19) > 10^11. - Giovanni Resta, Jun 04 2016

Examples

			765625' = 1137500; (1137500 + 1)' = 1137501' = 765624 = 765625 - 1, so 765625 is a term.
1137501' = 765624; (765624 + 1)' = 765625' = 1137500 = 1137501 - 1, so 1137501 is a term.
		

Crossrefs

Cf. A054377, A203617.

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);

Mathematica

A003415[n_]:=If[Abs[n]<2,0,n*Total[#2/#1&@@@FactorInteger[Abs[n]]]];
Select[Range[1,100000],A003415[A003415[#]+1]==#-1&] (* Julien Kluge, Jul 08 2016 *)

PARI

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

Extensions

a(17)-a(18) from Giovanni Resta, Jun 04 2016