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A175189
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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.
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0
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7, 33, 29, 59, 347, 2039, 4079, 32633, 65267, 913739, 1827479, 36549581
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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