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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 (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 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) : zz2 :=mul(zz1[i], 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

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