cp's OEIS Frontend

A359330 Composite k for which phi(k) + phi(k') = k, where k' is the arithmetic derivative of k (A003415).

%I A359330 #18 Feb 17 2023 22:05:35
%S A359330 4,6,8,10,12,18,22,28,34,58,60,72,82,84,88,108,112,118,124,132,140,
%T A359330 142,202,204,214,216,220,228,260,274,298,324,340,358,372,382,394,444,
%U A359330 454,478,492,508,538,562,564,572,580,620,622,644,694,708,740,804,812,820
%N A359330 Composite k for which phi(k) + phi(k') = k, where k' is the arithmetic derivative of k (A003415).
%C A359330 Composite numbers k for which phi(k') = cototient(k) (A051953).
%C A359330 The sequence refers only to composite numbers because for any prime number p we obtain phi(p) + phi(p') = p - 1 + phi(1) = p.
%C A359330 If p = 2^k - 1 is a Mersenne prime (A000668), then m = 4*p is a term. Indeed, m' = 4*(p + 1) = 4*2^k = 2^(k + 2) and phi(m) + phi(m') = phi(4*p) + phi(2^(k + 2)) = 2*(p-1) + 2^(k+1) = 2*(p - 1) + 2*(p + 1) = 4*p = m, so m is a term.
%C A359330 If p, q and p*q + p + q are prime numbers then m = 4*p*q is a term. Indeed, m'= 4*(p*q + p + q) and phi(m) + phi(m') = phi(4*p*q) + phi(4*(p*q + p + q)) = 2*(p - 1)*(q - 1) + 2*(p*q + p + q - 1) = 4*p*q.
%C A359330 If p is in A023221 then m = 20*p is a term. Indeed, m' = 24*p + 20 = 4*(6*p + 5) and phi(m) + phi(m') = phi(20*p) + phi(4*(6*p + 5)) = 8*(p-1) + 2*(6*p + 4) = 20*p = m, so m is a term.
%e A359330 If m = 4 then m' = 4 and phi(m) + phi(m') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
%e A359330 If m = 8 then m' = 12 and phi(m) + phi(m') = phi(8) + phi(12) = 4 + 4 = 8, so 8 is a term.
%e A359330 14 is not a term because phi(14) + phi(14') = 6 + phi(9) = 6 + 6 = 12 <> 14.
%p A359330 d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p A359330 q:= n-> not isprime(n) and (p-> p(n)+p(d(n))=n)(numtheory[phi]):
%p A359330 select(q, [$4..1000])[];  # _Alois P. Heinz_, Jan 29 2023
%t A359330 d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[1000], CompositeQ[#] && EulerPhi[#] + EulerPhi[d[#]] == # &] (* _Amiram Eldar_, Jan 29 2023 *)
%o A359330 (Magma) f:=func<h |h le 1 select 0 else h*(&+[Factorisation(h)[i][2] / Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>;  [n:n in [2..850]|not IsPrime(n) and n eq EulerPhi(Floor(f(n))) + EulerPhi(n)];
%Y A359330 Cf. A000010, A001359, A002808, A003415, A051953, A066938, A023221, A190402.
%K A359330 nonn
%O A359330 1,1
%A A359330 _Marius A. Burtea_, Jan 28 2023