login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A175189 Smallest integer m such that phi(phi(m))^n + tau(phi(m))^n = phi(rad(m))^n, where n is the number of iterations of phi(phi), tau(phi) and phi(rad) functions. 0
7, 33, 29, 59, 347, 2039, 4079, 32633, 65267, 913739, 1827479, 36549581 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Remarks about an interesting property of the equation phi(phi(m))^k + tau(phi(m))^k = phi(rad(m))^k: Let p be a prime number. If p is a solution of this equation with k iterations, and if q = 2*p+1 is prime, then q is solution of the equation with k+1 iterations.

Proof: we use the following properties, if p is prime: phi(phi(2*p+1)) = phi(2*p) = p-1; tau(phi(2*p+1)) = tau(2*p) = 4; phi(rad(2*p+1)) = phi(2*p+1) = 2*p; phi(phi(p)) = phi(p-1); tau(phi(p)) = tau(p-1); phi(rad(p)) = phi(p) = p-1.

Example: 2039 is prime and is solution for k = 6, and 4079 = 2*2039 + 1 is prime and is solution for n = 7; idem with the primes 32633, 913739, but p = 36549581 is prime and solution for 12 iterations, but 2*p + 1 is not prime, so it is not a solution.

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

LINKS

Table of n, a(n) for n=1..12.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

C. K. Caldwell, The Prime Glossary, Number of divisors

Wikipedia, Euler's totient function

EXAMPLE

For n=1, phi(phi(7)) = 2, tau(phi(7)) = 4, phi(rad(7)) = phi(7) = 6, then 2 + 4 = 6.

For n=2, phi(phi(phi(phi(33)))) = 2, tau(phi(tau(phi(33))) = 2, phi(rad(phi(rad(33) = 4, then 2 + 2 = 4.

For n=3, phi(phi(phi(phi(phi(phi(29)))))) = 1, tau(phi(tau(phi(tau(phi(29)))))) = 1, phi(rad(phi(rad(phi(rad(29)))))) = 2, then 1 + 1 = 2.

MAPLE

with(numtheory):for n from 1 to 100 do:indic:=0:for x from 1 to 10000 while(indic=0 ) do:x0:=x:y0:=x:z0:=x: for iter from 1 to n do:x1:=phi(phi(x0)): y1:= tau(phi(y0)): zz1:= ifactors(z0)[2] : zz2 :=mul(zz1[i][1], i=1..nops(zz1)): z1:=phi(zz2):x0:=x1:y0:=y1:z0:=z1:od :if x0 +y0=z0 then print (x):indic:=1:else fi:od:od:

PROG

(PARI) rad(m) = factorback(factorint(m)[, 1]); \\ A007947

phi_phi(m, n) = {for (k=1, n, m = eulerphi(eulerphi(m)); ); m; }

tau_phi(m, n) = {for (k=1, n, m = numdiv(eulerphi(m)); ); m; }

phi_rad(m, n) = {for (k=1, n, m = eulerphi(rad(m)); ); m; }

a(n) =  {my(m=1); while (phi_phi(m, n)+ tau_phi(m, n) != phi_rad(m, n), m++); m; } \\ Michel Marcus, Sep 17 2020

CROSSREFS

Cf. A000010 (phi), A007947 (rad), A000005 (tau), A002183.

Sequence in context: A110323 A288721 A324412 * A153286 A060745 A275163

Adjacent sequences:  A175186 A175187 A175188 * A175190 A175191 A175192

KEYWORD

nonn,more

AUTHOR

Michel Lagneau, Mar 01 2010

EXTENSIONS

Edited by Michel Marcus, Sep 17 2020

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 June 22 20:16 EDT 2021. Contains 345388 sequences. (Running on oeis4.)