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

1,2

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

Links

Table of n, a(n) for n=1..18.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A054377, A203617.

Sequence in context: A280043 A309813 A033936 * A334883 A098814 A272177

Adjacent sequences: A203615 A203616 A203617 * A203619 A203620 A203621

Keyword

nonn,more,changed

Author

Paolo P. Lava, Jan 20 2012

Extensions

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

Status

approved