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 A203618 Numbers m such that (m'+1)' = m-1, 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 (list; graph; refs; listen; history; text; internal format)
 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 couples 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 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 AUTHOR Paolo P. Lava, Jan 20 2012 EXTENSIONS a(17)-a(18) from Giovanni Resta, Jun 04 2016 STATUS approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)