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A218011 Numbers n for which n’ = x’*y’, where x>0, y>0, n = x + y and n’, x’, y’ are the arithmetic derivatives of n, x, y. 1
5, 7, 13, 19, 31, 43, 48, 55, 61, 73, 74, 87, 103, 106, 109, 117, 139, 146, 151, 159, 160, 178, 181, 193, 199, 202, 208, 212, 225, 229, 236, 241, 252, 267, 268, 271, 283, 285, 298, 313, 349, 357, 362, 386, 403, 411, 421, 433, 455, 463, 496, 511, 519, 523, 535 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The greatest prime in a twin primes couple is in the sequence. In fact if the twin primes are a and b, with a<b, b can be written as b=a+2. Being a’=b’=2’=1 we have b’=a’*2’ that is 1=1*1.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..250

EXAMPLE

n= 612, x=85,  y=527; n’=1056, x’=22, y’=48 and 1056=22*48.

n= 752, x=361, y=391; n’=1520, x’=38, y’=40 and 1520=38*40.

n= 779, x=36,  y=743; n’=60,   x’=60, y’=1  and 60=60*1.

MAPLE

with(numtheory);

A218011:= proc(i)

local a, b, c, n, p, pfs, q;

for n from 1 to i do

for q from 1 to trunc(n/2) do

  a:=q*add(op(2, p)/op(1, p), p= ifactors(q)[2]);

  b:=(n-q)*add(op(2, p)/op(1, p), p= ifactors(n-q)[2]);

  c:=n*add(op(2, p)/op(1, p), p= ifactors(n)[2]);

  if c=a*b then lprint(n, q, n-q); break; fi;

od; od;

end:

A218011(1000000);

MATHEMATICA

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; f[n_] := Select[Range[n/2], dn[#]*dn[n - #] == dn[n] &]; Select[Range[535], Length[f[#]] > 0 &] (* T. D. Noe, Oct 18 2012 *)

CROSSREFS

Cf. A003415, A211223, A211224, A211225, A212662, A212663, A212664.

Sequence in context: A243457 A189441 A106986 * A242255 A006512 A074304

Adjacent sequences:  A218008 A218009 A218010 * A218012 A218013 A218014

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Oct 18 2012

STATUS

approved

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Last modified June 19 17:15 EDT 2019. Contains 324222 sequences. (Running on oeis4.)