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%I #20 Aug 03 2017 04:16:02
%S 2,2,95,121,121,2047,49151,98303,393215,1572863,6291455,8388607,
%T 201326591,805306367,3221225471,6442450943,137438953471,137438953471,
%U 137438953471
%N Smallest number x > 1 such that phi(x) + sigma(x) = k*d(x)^n, i.e., the left-hand side is divisible by the n-th power of the number of divisors.
%C It appears that for n > 5, a(n) is a semiprime. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Aug 22 2005
%C a(20) <= 2199023255551. a(21) <= 2199023255551. - _Donovan Johnson_, Jun 05 2011
%F Least integer x > 1 such that A000010(x) + A000203(x) = k*A000005(x)^n.
%e The terms of list {2,2,95,121,121,2047,49151,98303} have {2,2,4,3,3,4,4,4} divisors, {3,3,120,133,133,2160,51312,99000} divisor-sums, {1,1,72,110,110,1936,46992,97608} EulerPhi values. The phi+sigma sums are {4,4,192,243,243,4096,98304,196608}, which are divisible by {2,4,64,81,243,4096,16384,65536} increasing powers of d-numbers, giving {2,1,3,3,1,1,6,3} quotients respectively.
%t Join[{2, 2}, Table[n = 3; While[! Divisible[(DivisorSigma[1, n] + EulerPhi[n]), DivisorSigma[0, n]^i], n += 2]; n, {i, 3, 12}]] (* _Jayanta Basu_, Jul 12 2013 *)
%o (PARI) k=2;for(n=1,15, while(denominator((sigma(k)+eulerphi(k))/(sigma(k,0)^n))!=1,k++);\ print(n," ",k)) (Klasen)
%Y Cf. A000005, A000010, A000203.
%K nonn
%O 1,1
%A _Labos Elemer_, Jun 27 2000
%E More terms from _Jud McCranie_, Oct 08 2000
%E More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Aug 22 2005
%E a(16)-a(19) from _Donovan Johnson_, Jun 05 2011