A053650 Cototient function of n^2.

0, 2, 3, 8, 5, 24, 7, 32, 27, 60, 11, 96, 13, 112, 105, 128, 17, 216, 19, 240, 189, 264, 23, 384, 125, 364, 243, 448, 29, 660, 31, 512, 429, 612, 385, 864, 37, 760, 585, 960, 41, 1260, 43, 1056, 945, 1104, 47, 1536, 343, 1500, 969, 1456, 53, 1944, 825, 1792, 1197

Offset

1,2

Comments

Seems to be invertible like n*Phi(n). Compare with A002618, A038040.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = n*(n - phi(n)) = n^2 - n*phi(n) = Cototient(n^2) = A051953(A000290(n)).

a(n) = n^2 - A002618(n).

For p prime, Cototient(p)=1 and a(p)=p.

a(n) = n*cototient(n) = n*A051953(n). - Omar E. Pol, Nov 22 2012

Dirichlet g.f.: zeta(s-2)*(1 - 1/zeta(s-1)). - Ilya Gutkovskiy, Jul 26 2016

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023

Mathematica

Table[n(n-EulerPhi[n]), {n, 60}] (* Michael De Vlieger, Jul 26 2016 *)

Prog

(PARI) a(n) = n^2 - eulerphi(n^2) \\ Michel Marcus, Jul 27 2013
(Haskell)
a053650 = a051953 . a000290  -- Reinhard Zumkeller, Jan 21 2014
(Magma) [n*(n-EulerPhi(n)): n in [1..60]]; // Vincenzo Librandi, Jul 27 2016
(Sage) [n*(n - euler_phi(n)) for n in (1..60)] # G. C. Greubel, May 18 2019
(GAP) List([1..60], n-> n*(n- Phi(n)) ); # G. C. Greubel, May 18 2019

Crossrefs

Cf. A000005, A038040.

Cf. A001248, A002618, A053650, A053192, A053193, A036689, A229099.

Sequence in context: A062956 A333375 A289667 * A377486 A119794 A117987

Adjacent sequences: A053647 A053648 A053649 * A053651 A053652 A053653

Keyword

nonn,look,easy

Author

Labos Elemer, Feb 18 2000

Status

approved