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A165562 Numbers n for which n+n' is prime, n' being the arithmetic derivative of n.

%I A165562 #15 Feb 04 2022 10:36:59
%S A165562 2,6,10,14,15,21,26,30,33,34,35,38,42,46,51,55,57,58,65,66,74,78,85,
%T A165562 86,93,102,110,111,118,123,141,143,145,155,158,161,166,177,178,182,
%U A165562 185,186,194,201,203,205,206,209,210,215,221,230,246,254,258,267,278,282,290
%N A165562 Numbers n for which n+n' is prime, n' being the arithmetic derivative of n.
%C A165562 The only prime in this sequence is 2. Since it is the only even prime and p' = 1, it is the only prime that added to its derivative can give an odd prime (namely 3).
%H A165562 T. D. Noe, <a href="/A165562/b165562.txt">Table of n, a(n) for n = 1..1000</a>
%e A165562 46 is in the list because: n=46 -> n'=25 -> n+n'=71 that is prime.
%p A165562 with(numtheory);
%p A165562 P:= proc(n)
%p A165562 local a,i,p,pfs;
%p A165562 for i from 1 to n do
%p A165562 pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs);
%p A165562 if isprime(a+i) then print(i); fi;
%p A165562 od;
%p A165562 end:
%p A165562 P(1000);
%p A165562 # alternative
%p A165562 isA165562 := proc(n)
%p A165562     isprime(A129283(n)) ;
%p A165562 end proc:
%p A165562 for n from 1 to 1000 do
%p A165562     if isA165562(n) then
%p A165562         printf("%d,",n) ;
%p A165562     end if;
%p A165562 end do: # _R. J. Mathar_, Feb 04 2022
%t A165562 (*First run the program given in A003415*) A165562 = Select[ Range[ 1000 ], PrimeQ[ # + a[ # ] ] & ]
%o A165562 (Python)
%o A165562 from sympy import isprime, factorint
%o A165562 A165562 = [n for n in range(1,10**5) if isprime(n+sum([int(n*e/p) for p,e in factorint(n).items()]))] # _Chai Wah Wu_, Aug 21 2014
%Y A165562 Cf. A003415, A165561.
%K A165562 easy,nonn
%O A165562 1,1
%A A165562 _Paolo P. Lava_ and _Giorgio Balzarotti_, Sep 25 2009
%E A165562 Terms verified by _Alonso del Arte_, Oct 30 2009