A165558 Integers that are half of their arithmetic derivatives.

0, 16, 108, 729, 12500, 84375, 3294172, 9765625, 22235661, 2573571875, 678223072849, 1141246682444, 7703415106497, 891598970659375, 1211500426369012, 8177627877990831, 234966429149994773, 946484708100790625

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 22 2009

Keywords

• nonn

Comments

All integers of the form p^p*q^q, with q and p two distinct primes, are in the sequence. [R. J. Mathar, Sep 26 2009]

6*10^8 < a(10) <= 2573571875. a(11) <= 678223072849. [Donovan Johnson, Nov 03 2010]

By a result of Ufnarovski and Ahlander, an integer is in this sequence if and only if it has the form p^(2p) or p^p*q^q, with p and q distinct primes. See comments from A072873. [Nathaniel Johnston, Nov 22 2010]

Examples

			For k =84375 = 3^3*5^5, so A003415(k)/2 = 84375*(3/3+5/5)/2 = 84375 = k, which adds k=84375 to the sequence.
		

Crossrefs

Cf. A003415, A072873.

Programs

Maple

with(numtheory);
P:=proc(n)
local a,i,p,pfs;
for i from 1 to n do
  pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs);  if a=2*i then print(i); fi; od;
end:
P(100000000);

Mathematica

d[0] = d[1] = 0; d[n_] := n*Total[f = FactorInteger[n]; f[[All, 2]]/f[[All, 1]] ]; Join[{0}, Reap[Do[p = Prime[n]; ip = p^(2*p); If[ip == d[ip]/2, Sow[ip]]; Do[q = Prime[k]; iq = p^p*q^q; If[iq == d[iq]/2, Sow[iq]], {k, n+1, 6}], {n, 1, 5}]][[2, 1]] // Union][[1 ;; 18]] (* Jean-François Alcover, Apr 22 2015, after Nathaniel Johnston *)

Formula

{n: A003415(n) = 2*n}.

Extensions

Entries checked by R. J. Mathar, Sep 26 2009

a(7)-a(9) from Donovan Johnson, Nov 03 2010

a(10)-a(18) and general form from Nathaniel Johnston, Nov 22 2010