A092693 Sum of iterated phi(n).

0, 1, 3, 3, 7, 3, 9, 7, 9, 7, 17, 7, 19, 9, 15, 15, 31, 9, 27, 15, 19, 17, 39, 15, 35, 19, 27, 19, 47, 15, 45, 31, 35, 31, 39, 19, 55, 27, 39, 31, 71, 19, 61, 35, 39, 39, 85, 31, 61, 35, 63, 39, 91, 27, 71, 39, 55, 47, 105, 31, 91, 45, 55, 63, 79, 35, 101, 63, 79, 39, 109, 39, 111

Offset

1,3

Comments

Iannucci, Moujie and Cohen examine perfect totient numbers: n such that a(n) = n.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

C. Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015) # 15.2.1

P. Erdos and M. V. Subbarao, On the iterates of some arithmetic functions, The theory of arithmetic functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich. 1971), Lecture Notes in Math., 251 , pp. 119-125, Springer, Berlin, 1972. [alternate link]

Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On perfect totient numbers, J. Integer Sequences, 6 (2003), #03.4.5.

Formula

a(1) = 0, a(n) = phi(n) + a(phi(n))

a(n) = A053478(n) - n. - Vladeta Jovovic, Jul 02 2004

Erdős & Subbarao prove that a(n) ~ phi(n) for almost all n. In particular, a(n) < n for almost all n. The proportion of numbers up to N for which a(n) > n is at most 1/log log log log N. - Charles R Greathouse IV, Mar 22 2012

Example

a(100) = 71 because the iterations of phi (40, 16, 8, 4, 2, 1) sum to 71.

Mathematica

nMax=100; a=Table[0, {nMax}]; Do[e=EulerPhi[n]; a[[n]]=e+a[[e]], {n, 2, nMax}]; a (* T. D. Noe *)
Table[Plus @@ FixedPointList[EulerPhi, n] - (n + 1), {n, 72}] (* Alonso del Arte, Jan 29 2007 *)

Prog

(Haskell)
a092693 1 = 0
a092693 n = (+ 1) $ sum $ takeWhile (/= 1) $ iterate a000010 $ a000010 n
-- Reinhard Zumkeller, Oct 27 2011
(PARI) a(n)=my(k); while(n>1, k+=n=eulerphi(n)); k \\ Charles R Greathouse IV, Mar 22 2012
(Python)
from sympy import totient
from math import prod
def f(n):
    m = n
    while m > 1:
        m = totient(m)
        yield m
def A092693(n): return sum(f(n)) # Chai Wah Wu, Nov 14 2021

Crossrefs

Cf. A003434 (iterations of phi(n) needed to reach 1), A092694 (iterated phi product).

Cf. A082897 and A091847 (perfect totient numbers).

Sequence in context: A145501 A370296 A182139 * A134661 A135434 A204204

Adjacent sequences: A092690 A092691 A092692 * A092694 A092695 A092696

Keyword

nonn

Author

T. D. Noe, Mar 04 2004

Status

approved