A257107 Composite numbers n such that n'=(n+12)', where n' is the arithmetic derivative of n.

16, 65, 88, 209, 11009, 38009, 680609, 2205209, 2860198, 3515609, 4347209, 5365387, 5809361, 10595009, 12006209, 31979009, 83255059, 89019209, 152915402, 169130009, 172147423, 225869899, 244766009, 247590209, 258084209, 325622009, 357777209, 377330609

Offset: 1

Author

Paolo P. Lava, Apr 17 2015

Keywords

• nonn

Comments

If the limitation of being composite is removed we also have the numbers p such that if p is prime then p + 12 is prime too (A046133).

Examples

			16' = (16 + 12)' = 28' = 32;
65' = (65 + 12)' = 77' = 18.
		

Crossrefs

Cf. A003415, A046133, A226779.

Programs

Maple

with(numtheory); P:= proc(q,h) local a,b,n,p;
for n from 1 to q do if not isprime(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=(n+h)*add(op(2,p)/op(1,p),p=ifactors(n+h)[2]);
if a=b then print(n); fi; fi; od; end: P(10^9,12);

Mathematica

a[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]];
Select[Range@ 100000, And[CompositeQ@ #, a@# == a[# + 12]] &] (* Michael De Vlieger, Apr 22 2015, after Michael Somos at A003415 *)

Extensions

a(16)-a(28) from Lars Blomberg, May 06 2015