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.

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

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

Antti Karttunen, Table of n, a(n) for n = 1..12822 (first 250 terms from Paolo P. Lava)

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 *)

Prog

(PARI)
up_to = 2^18;
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v003415 = vector(up_to, n, A003415(n));
isA218011(n) = { my(z=v003415[n]); for(x=2, ceil(n/2), if(!(z%v003415[x]), if(z==v003415[x]*v003415[n-x], return(1)))); (0); }; \\ Antti Karttunen, Feb 22 2024

Crossrefs

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

Subsequences: A006512 (primes in this sequence), A370126 (k with a solution where both x and y are composite).

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