login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057635 a(n) is the largest m such that phi(m) = n, where phi is Euler's totient function = A000010. 12
2, 6, 0, 12, 0, 18, 0, 30, 0, 22, 0, 42, 0, 0, 0, 60, 0, 54, 0, 66, 0, 46, 0, 90, 0, 0, 0, 58, 0, 62, 0, 120, 0, 0, 0, 126, 0, 0, 0, 150, 0, 98, 0, 138, 0, 94, 0, 210, 0, 0, 0, 106, 0, 162, 0, 174, 0, 118, 0, 198, 0, 0, 0, 240, 0, 134, 0, 0, 0, 142, 0, 270, 0, 0, 0, 0, 0, 158, 0, 330, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

To check that a property P holds for all EulerPhi(x) not exceeding n, for n with a(n) > 0, it suffices to check P for all EulerPhi(x) with x not exceeding a(n). - Joseph L. Pe, Jan 10 2002

The Alekseyev link in A131883 establishes the following explicit relationship between A131883, A036912 and A057635. Namely, for t belonging to A036912, we have t = A131883(A057635(t)-1). In other words, A036912(n) = A131883(A057635(A036912(n))-1) for all n.

From Jianing Song, Feb 16 2019: (Start)

Let f(n) = exp(gamma)*log(log(n)) + 2.5/log(log(n)), then a(n) < n*f(n^2) for all n > 1, where gamma = A001620.

Proof. Without loss of generality we suppose log(log(n)) > n_0 = sqrt(2.5/exp(gamma)) = 1.18475..., then f(n), n/f(n) and N(n) = ceiling(n*f(n^2)) are all monotonically increasing functions of n, and we have f(n) < 2*exp(gamma)*log(log(n)).

By the formula (3.41) in Theorem 15 by J. Barkley Rosser and Lowell Schoenfeld we have phi(k) > k/f(k) for k != 1, 2, 223092870. N(31802157) = 223092869 < 223092870, N(31802158) = 223092877 > 223092870, so N(n) != 223092870 (N(n) is increasing). So phi(N(n)) > N(n)/f(N(n)) > (n*f(n^2))/f(n*f(n^2)) (n/f(n) is increasing and log(log(n*f(n^2))) > n_0).

Note that f(n^2) < 2*exp(gamma)*log(log(n^2)) < 2*exp(gamma)*(log(n^2)/e) = 4*exp(gamma-1)*log(n) < 4*exp(gamma-2)*n < n, so n*f(n^2) < n^2, f(n*f(n^2)) < f(n^2) (f(n) is increasing and log(log(n*f(n^2))) > n_0), so phi(N(n)) > n. As a result, a(n) <= N(n) - 1 < n*f(n^2).

Conjecturally a(n) < n*f(n) for all n > 2. (End)

LINKS

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

Rosser, J. Barkley; Schoenfeld, Lowell (1962). Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1): 64-94.

FORMULA

a(2n+1) = 0 for n > 0 and when a(2n) = 0, the nontotients (A005277)/2.

EXAMPLE

m = 12 is the largest value of m such that phi(m) = 4, so a(4) = 12.

MATHEMATICA

a = Table[0, {100}]; Do[ t = EulerPhi[n]; If[t < 101, a[[t]] = n], {n, 1, 10^6}]; a

PROG

(PARI) a(n) = if(n%2, 2*(n==1), forstep(k=floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))), n, -1, if(eulerphi(k)==n, return(k)); if(k==n, return(0)))) \\ Jianing Song, Feb 15 2019

CROSSREFS

Cf. A000010, A014197.

Sequence in context: A065344 A131105 A321713 * A269943 A243015 A139717

Adjacent sequences:  A057632 A057633 A057634 * A057636 A057637 A057638

KEYWORD

nonn

AUTHOR

Jud McCranie, Oct 10 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 08:20 EST 2019. Contains 329877 sequences. (Running on oeis4.)