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A362140 Numbers k in A224486 for which the arithmetic derivative k' (A003415) is also in A224486.

%I A362140 #11 Jul 04 2023 09:29:14
%S A362140 2,5,6,9,14,18,29,33,41,53,54,65,69,89,113,134,141,158,173,198,209,
%T A362140 221,233,249,278,281,293,326,329,338,393,473,506,509,545,581,593,614,
%U A362140 629,641,653,713,729,749,761,809,846,905,950,953,965,986,1013,1014,1026,1041,1049
%N A362140 Numbers k in A224486 for which the arithmetic derivative k' (A003415) is also in A224486.
%C A362140 Sophie Germain primes p that are not Lucasian primes (A103579) are terms because p' = 1 = A224486(1).
%e A362140 6 = A224486(4) and 6' = 5 = A224486(3), so 6 is a term.
%e A362140 9 = A224486(5) and 9' = 6 = A224486(4), so 9 is a term.
%e A362140 14 = A224486(6) and 14' = 9 = A224486(5), so 14 is a term.
%t A362140 d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); curzonQ[n_] := PowerMod[2, n, 2*n + 1] == 2*n; Select[Range[2, 1050], curzonQ[#] && curzonQ[d[#]] &] (* _Amiram Eldar_, May 05 2023 *)
%o A362140 (Magma) czn:=func<n|(2^n+1) mod (2*n+1) eq 0>; f:=func<n |n le 1 select 0 else
%o A362140 n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in
%o A362140 [1..#Factorisation(n)]])>; [n:n in [2..5000]|czn(n) and czn(Floor(f(n)))];
%Y A362140 Cf. A003415, A103579, A224486.
%K A362140 nonn
%O A362140 1,1
%A A362140 _Marius A. Burtea_, May 03 2023