A053158 - OEIS

%I #25 Feb 12 2024 01:55:34

%S 1,3,4,6,6,10,8,12,12,16,12,20,14,22,22,24,18,30,20,32,30,34,24,40,30,

%T 40,36,44,30,52,32,48,46,52,46,60,38,58,54,64,42,72,44,68,66,70,48,80,

%U 56,80,70,80,54,90,70,88,78,88,60,104,62,94,90,96,82,112,68,104,94,116

%N Sum of n and its cototient function value (A051953): a(n) = 2*n - phi(n), where phi is Euler phi.

%C For Mersenne primes and also for certain composites the values of this function are powers of 2.

%H Antti Karttunen, <a href="/A053158/b053158.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = n + A051953(n) = 2n - phi(n), where phi is A000010.

%F a(2^k) = 3*2^(k-1).

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 1 - 3/Pi^2 = 0.696036... . - _Amiram Eldar_, Dec 16 2023

%e a(127) = 254 - 126 = 128.

%e a(80) = 160 - 32 = 128.

%t a[n_] := 2*n - EulerPhi[n]; Array[a, 60] (* _Amiram Eldar_, Dec 16 2023 *)

%o (PARI) a(n) = 2*n - eulerphi(n); \\ _Michel Marcus_, Dec 19 2013

%o (Magma) [2*n - EulerPhi(n): n in [1..100]]; // _G. C. Greubel_, Feb 12 2024

%o (SageMath) [2*n - euler_phi(n) for n in range(1,101)] # _G. C. Greubel_, Feb 12 2024

%Y Cf. A000010, A000043, A000668, A001368, A051953, A104141.

%K nonn,easy

%O 1,2

%A _Labos Elemer_, Feb 29 2000

%E Name amended with formula by _Antti Karttunen_, Nov 15 2021