

A203618


Numbers m such that (m'+1)' = m1, where m' is the arithmetic derivative of m.


3



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

1,2


COMMENTS

The differential equation whose solutions are the primary pseudoperfect numbers is m' = k*m1, 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*m1) = (k'+k^2)*mk. Let k=1: (m'+1)' = m1. The solutions of this equation are the primary pseudoperfect numbers plus couples of numbers (x,y) for which x' = y1 and y' = x1.
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=n1 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) == n1; \\ 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


AUTHOR

Paolo P. Lava, Jan 20 2012


EXTENSIONS

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


STATUS

approved



