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%I #16 Mar 16 2019 19:54:28
%S 16,65,88,209,11009,38009,680609,2205209,2860198,3515609,4347209,
%T 5365387,5809361,10595009,12006209,31979009,83255059,89019209,
%U 152915402,169130009,172147423,225869899,244766009,247590209,258084209,325622009,357777209,377330609
%N Composite numbers n such that n'=(n+12)', where n' is the arithmetic derivative of n.
%C 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).
%e 16' = (16 + 12)' = 28' = 32;
%e 65' = (65 + 12)' = 77' = 18.
%p with(numtheory); P:= proc(q,h) local a,b,n,p;
%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]);
%p if a=b then print(n); fi; fi; od; end: P(10^9,12);
%t a[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]];
%t Select[Range@ 100000, And[CompositeQ@ #, a@# == a[# + 12]] &] (* _Michael De Vlieger_, Apr 22 2015, after _Michael Somos_ at A003415 *)
%Y Cf. A003415, A046133, A226779.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Apr 17 2015
%E a(16)-a(28) from _Lars Blomberg_, May 06 2015
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