cp's OEIS Frontend

A352332 Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).

%I A352332 #24 Dec 26 2024 08:32:39
%S A352332 4,260,294,740,1460,3140,3860,5540,8420,10820,15140,19940,21860,24020,
%T A352332 24260,27620,37460,40340,46820,49460,55940,61220,70340,85460,101540,
%U A352332 114020,124340,132740,133220,144260,148340,149540,155060,162020,164420,167060,170420,173540
%N A352332 Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).
%C A352332 If p > 3 is at the intersection of A023221 and A005383, then m = 20*p is a term. Indeed, m' = (20*p)' = 24*p + 20 = 4*(6*p + 5), m'' = (4*(6*p + 5))' = 4*(6*p + 6) = 24*(p + 1), phi(m') + phi(m'') = phi(4*(6*p + 5)) + phi(24*(p + 1)) = 2*(6*p + 4) + phi(48*(p + 1)/2) = 2*(6*p + 4) + 16*(p - 1)/2 = 12*p + 8 + 8*p - 8 = 20*p = m.
%e A352332 phi(4') + phi(4'') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
%e A352332 phi(260') + phi(260'') = phi(332) + phi(336) = 164 + 96 = 260, so 260 is a term.
%t A352332 d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200000], CompositeQ[#] && EulerPhi[d[#]] + EulerPhi[d[d[#]]] == # &] (* _Amiram Eldar_, Apr 10 2022 *)
%o A352332 (Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [n:n in [2..174000]|not IsPrime(n) and n-EulerPhi(Floor(f(n))) eq EulerPhi(Floor(f(Floor(f(n)))))];
%o A352332 (PARI)
%o A352332 ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A352332 isok(k) = my(adk=ad(k)); !isprime(k) && (k == eulerphi(adk) + eulerphi(ad(adk))); \\ _Michel Marcus_, Apr 30 2022
%Y A352332 Cf. A000010, A023221, A003415, A005383, A068346.
%K A352332 nonn
%O A352332 1,1
%A A352332 _Marius A. Burtea_, Apr 09 2022