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

4, 6, 8, 10, 12, 18, 22, 28, 34, 58, 60, 72, 82, 84, 88, 108, 112, 118, 124, 132, 140, 142, 202, 204, 214, 216, 220, 228, 260, 274, 298, 324, 340, 358, 372, 382, 394, 444, 454, 478, 492, 508, 538, 562, 564, 572, 580, 620, 622, 644, 694, 708, 740, 804, 812, 820

Offset

1,1

Comments

Composite numbers k for which phi(k') = cototient(k) (A051953).

The sequence refers only to composite numbers because for any prime number p we obtain phi(p) + phi(p') = p - 1 + phi(1) = p.

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.

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.

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.

Links

Table of n, a(n) for n=1..56.

Example

If m = 4 then m' = 4 and phi(m) + phi(m') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
If m = 8 then m' = 12 and phi(m) + phi(m') = phi(8) + phi(12) = 4 + 4 = 8, so 8 is a term.
14 is not a term because phi(14) + phi(14') = 6 + phi(9) = 6 + 6 = 12 <> 14.

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> not isprime(n) and (p-> p(n)+p(d(n))=n)(numtheory[phi]):
select(q, [$4..1000])[];  # Alois P. Heinz, Jan 29 2023

Mathematica

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 *)

Prog

(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)];

Crossrefs

Cf. A000010, A001359, A002808, A003415, A051953, A066938, A023221, A190402.

Sequence in context: A341280 A034288 A131984 * A307542 A225510 A131694

Adjacent sequences: A359327 A359328 A359329 * A359331 A359332 A359333

Keyword

nonn

Author

Marius A. Burtea, Jan 28 2023

Status

approved