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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
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
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].
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: A288721 A324412 A373442 * A153286 A060745 A275163
KEYWORD
nonn,more
AUTHOR
Michel Lagneau, Mar 01 2010
EXTENSIONS
Edited by Michel Marcus, Sep 17 2020
STATUS
approved