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

1,2

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]

Links

Table of n, a(n) for n=1..18.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A003415, A072873.

Sequence in context: A056001 A163725 A269188 * A337391 A250425 A238171

Adjacent sequences: A165555 A165556 A165557 * A165559 A165560 A165561

Keyword

nonn

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 22 2009

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

Status

approved