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).

4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620, 37460, 40340, 46820, 49460, 55940, 61220, 70340, 85460, 101540, 114020, 124340, 132740, 133220, 144260, 148340, 149540, 155060, 162020, 164420, 167060, 170420, 173540

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..38.

Example

phi(4') + phi(4'') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
phi(260') + phi(260'') = phi(332) + phi(336) = 164 + 96 = 260, so 260 is a term.

Mathematica

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 *)

Prog

(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)))))];
(PARI)
ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = my(adk=ad(k)); !isprime(k) && (k == eulerphi(adk) + eulerphi(ad(adk))); \\ Michel Marcus, Apr 30 2022

Crossrefs

Cf. A000010, A023221, A003415, A005383, A068346.

Sequence in context: A116967 A003382 A112982 * A061788 A203839 A052136

Adjacent sequences: A352329 A352330 A352331 * A352333 A352334 A352335

Keyword

nonn

Author

Marius A. Burtea, Apr 09 2022

Status

approved