A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

4, 27, 104, 260, 296, 405, 525, 740, 910, 945, 1460, 1806, 1818, 2504, 3125, 3140, 3176, 3656, 3860, 4563, 5540, 6056, 6930, 7016, 8420, 8636, 9224, 10820, 12573, 13256, 14024, 15140, 15464, 15944, 16136, 19940, 20456, 21690, 21860, 22856, 23336, 24020, 24260

Offset: 1

Author

Marius A. Burtea, Apr 09 2022

Keywords

• nonn

Comments

If m is a term in A051674, then m'' = m, phi(m'') = phi(m) so the sequence is infinite.

If p > 3 is at the intersection of A023208 and A005383 then m = 8*p is a term. Indeed, m'' = (8*p)'' = (12*p + 8)' = (4*(3*p + 2))' = 12*(p + 1) and phi(m'') = phi(12*(p + 1)) = phi(24*(p + 1)/2) = 8*(p - 1)/2 = 4*(p - 1) and phi(m) = phi(8*p) = 4*(p - 1).

If p > 5 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))' = 4*(6*p + 6) = 24*(p + 1) and phi(m'') = phi(24*(p + 1)) = phi(48*(p + 1)/2) = 16*(p - 1)/2 = 8*(p - 1) and phi(m) = phi(20*p) = 8*(p - 1).

Examples

			phi(4'') = phi(4) because 4'' = 4, so 4 is a term.
phi (27'') = phi(27) because 27'' = 27, so 27 is a term.
phi(104'') = phi(164') = phi(168) = phi (8*3*7) = 4*2*6 = 48 and phi(104) = phi(8*13) = 4*12 = 48, so 104 is a term.
		

Crossrefs

Cf. A000010, A005383, A051674, A068346, A023208, A023221, A190402,

Programs

Magma

f:=func; [n:n in [2..24300]| not IsPrime(n) and EulerPhi(n) eq EulerPhi(Floor(f(Floor(f(n))))) ];

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[25000], EulerPhi[#] == EulerPhi[d[d[#]]] &] (* Amiram Eldar, Apr 10 2022 *)