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A165561 Primes that are the sum of an integer n and its arithmetic derivative.

%I A165561 #14 Jul 22 2025 07:23:36
%S A165561 3,11,17,23,31,41,47,53,59,61,71,79,83,89,107,113,127,131,149,151,167,
%T A165561 179,191,193,197,227,239,251,263,269,271,293,311,313,347,359,383,401,
%U A165561 419,431,439,443,449,457,479,491,503,521,523,587,593,599,607,617,631
%N A165561 Primes that are the sum of an integer n and its arithmetic derivative.
%C A165561 Some primes are the sum of an integer and its derivative in more than one way (e.g., 23, 71, 191 (not a complete listing within the range shown)). Just calculating this sequence from A165562 gives a list that is not sorted in ascending order and contains duplicate items. However, since in the range from 1 to 10000 only the number 1 and the primes have arithmetic derivatives that are less than their square roots, I feel confident that the list given above is not missing some term that corresponds to a large value in A165562. In other words, for a term to be missing from the list above, its corresponding value in A165562 would have to be less than 625. - _Alonso del Arte_, Oct 30 2009
%H A165561 T. D. Noe, <a href="/A165561/b165561.txt">Table of n, a(n) for n = 1..1000</a>
%F A165561 {p in A000040: p in A129283}. - _R. J. Mathar_, Feb 04 2022
%e A165561 71 is in the list because: n=46 -> n'=25 -> n+n'=71; n=51 -> n'=20 -> n+n'=71; n=55 -> n'=16 -> n+n'=71.
%p A165561 isA165561 := proc(n)
%p A165561     if isprime(n) then
%p A165561         for i from 1 to n do
%p A165561             if n = A129283(i) then
%p A165561                 return true ;
%p A165561             end if;
%p A165561         end do:
%p A165561         false ;
%p A165561     else
%p A165561         false;
%p A165561     end if;
%p A165561 end proc:
%p A165561 for n from 2 to 1000 do
%p A165561     if isA165561(n) then
%p A165561         printf("%d,",n) ;
%p A165561     end if;
%p A165561 end do: # _R. J. Mathar_, Feb 04 2022
%t A165561 (*First run the program given in A165562*) SetAttributes[a, Listable]; A165561 = Union[A165562 + a[A165562]]
%Y A165561 Cf. A003415, A165562
%K A165561 easy,nonn
%O A165561 1,1
%A A165561 _Paolo P. Lava_ and _Giorgio Balzarotti_, Sep 25 2009
%E A165561 Terms verified by _Alonso del Arte_, Oct 30 2009