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%I #12 Sep 08 2022 08:46:15
%S 5,7,11,13,17,73,257,2593,65537
%N Primes p such that p-1 = phi(p) = k*phi(p-k) for some number 1 <= k < p-1.
%C Corresponding values of numbers k: 2, 3, 5, 3, 2, 3, 2, 3, 2, ...
%C 83623937 is also a term of this sequence (with k = 2).
%C For all primes p we have: phi(p) = k*phi(p-k) if k = p - 1.
%C Primes from A266267.
%C The first 4 known Fermat primes > 3 from A019434 are in sequence.
%e 17 is in the sequence because phi(17) = 16 = 2*phi(15) = 2*8.
%t Select[Prime@ Range@ 500, Function[p, AnyTrue[Range[p - 2], p - 1 == # EulerPhi[p - #] &]]] (* _Michael De Vlieger_, Jan 09 2016, Version 10 *)
%o (Magma) Set(Sort([[n: k in [1..n-2] | IsPrime(n) and EulerPhi(n) eq k*EulerPhi(n-k)]: n in [1..10000]]))
%o (Magma) Set(Sort([5] cat [n: n in [6..100000], k in [1..5] | IsPrime(n) and EulerPhi(n) eq k*EulerPhi(n-k)]))
%o (PARI) listp(nn) = {forprime(p=2, nn, for (k=1, p-2, if (eulerphi(p) == k*eulerphi(p-k), print1(p, ", "); break)););} \\ _Michel Marcus_, Dec 27 2015
%Y Cf. A000010, A019434, A266267.
%K nonn,more
%O 1,1
%A _Jaroslav Krizek_, Dec 26 2015
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