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%I #28 Sep 30 2023 10:50:54
%S 3,4,5,17,257,65537,83623937
%N Numbers k such that k = tau(k) * phi(k-2) + 1.
%C Numbers k such that k = A000005(k) * A000010(k-2) + 1.
%C Sequence deviates from A249541; numbers 4294967297 and 6992962672132097 are not terms of this sequence.
%C The first 5 known Fermat primes from A019434 are in this sequence.
%C Conjecture: primes from this sequence are in A254576.
%C a(8) > 10^13. If k = tau(k) * phi(k-2) + 1 then phi(k-2) must divide k-1, thus k-2 must be a term of A203966, which has already been searched up to 10^13. - _Giovanni Resta_, Feb 21 2020
%C a(8) > 10^17. - _Max Alekseyev_, Sep 29 2023
%e 17 is in this sequence because 17 = tau(17)*phi(15) + 1 = 2*8 + 1.
%t Select[Range@ 100000, # == DivisorSigma[0, #] EulerPhi[# - 2] + 1 &] (* _Michael De Vlieger_, Oct 27 2015 *)
%o (Magma) [n: n in [3..1000000] | n eq NumberOfDivisors(n) * EulerPhi(n-2) + 1]
%o (PARI) for(n=3, 1e8, if(numdiv(n)*eulerphi(n-2) == n-1, print1(n ", "))) \\ _Altug Alkan_, Oct 28 2015
%o (PARI) lista(na, nb) = {my(f1 = factor(na-2), f2 = factor(na-1), f3); for(n=na, nb, f3 = factor(n); if (numdiv(f3)*eulerphi(f1) == n-1, print1(n ", ")); f1 = f2; f2 = f3;);}; \\ _Michel Marcus_, Feb 21 2020
%Y Cf. A000005, A000010, A019434, A249541, A254576, A203966.
%Y Cf. A263811 (numbers k such that k = tau(k) * phi(k-1) + 1).
%K nonn,more
%O 1,1
%A _Jaroslav Krizek_, Oct 27 2015