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A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

%I A240960 #21 Mar 08 2019 07:23:08
%S A240960 2,3,5,7,11,13,15,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A240960 89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,
%U A240960 179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269
%N A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.
%C A240960 a(n) = A182140(n) for n <= 35.
%C A240960 All primes p are in the sequence since (p+1) - (p-1) = 2^1. The first composites are 15, 119748396, 139254850, 187768485, 1420027536, 3991789984. A182140 seems unrelated. - _Jens Kruse Andersen_, Aug 05 2014
%H A240960 Reinhard Zumkeller, <a href="/A240960/b240960.txt">Table of n, a(n) for n = 1..1000</a>
%p A240960 with(numtheory):
%p A240960 filter:= n -> sigma(n) - phi(n) = tau(n)^nops(factorset(n)):
%p A240960 select(filter, [$1..1000]); # _Robert Israel_, Aug 05 2014
%t A240960 Select[Range[300], DivisorSigma[1, #] - EulerPhi[#] == DivisorSigma[0, #]^PrimeNu[#]&] (* _Jean-François Alcover_, Mar 08 2019 *)
%o A240960 (Haskell)
%o A240960 a240960 n = a240960_list !! (n-1)
%o A240960 a240960_list = filter (\x -> a051612 x == a110088 x) [1..]
%o A240960 (Python)
%o A240960 from sympy import totient,divisors,divisor_count,primefactors
%o A240960 filter(lambda x:sum(divisors(x))-totient(x)==divisor_count(x)**len(primefactors(x)), range(1,10**5)) # _Chai Wah Wu_, Aug 05 2014
%o A240960 (PARI) is(n)=my(f=factor(n)); sigma(f)-eulerphi(f)==numdiv(f)^omega(f) \\ _Charles R Greathouse IV_, Nov 26 2014
%Y A240960 Cf. A000005, A000010, A000203, A001221, A051612, A110088, A182140, A110086.
%K A240960 nonn
%O A240960 1,1
%A A240960 _Reinhard Zumkeller_, Aug 05 2014