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A263653 a(n) = bigomega(n)^omega(n).

%I A263653 #24 Feb 16 2025 08:33:27
%S A263653 1,1,2,1,4,1,3,2,4,1,9,1,4,4,4,1,9,1,9,4,4,1,16,2,4,3,9,1,27,1,5,4,4,
%T A263653 4,16,1,4,4,16,1,27,1,9,9,4,1,25,2,9,4,9,1,16,4,16,4,4,1,64,1,4,9,6,4,
%U A263653 27,1,9,4,27,1,25,1,4,9,9,4,27,1,25,4,4,1,64,4,4,4,16,1,64,4,9,4,4,4,36,1,9,9,16,1,27,1,16,27,4,1,25,1,27,4,25,1,27,4,9,9,4,4,125
%N A263653 a(n) = bigomega(n)^omega(n).
%C A263653 a(n) = 1 if n is prime (A000040), a(n) > 1 if n is composite (A002808), a(n) = 2 if n is the square of a prime (A001248), a(n) = 3 if n is the cube of a prime (A030078).
%C A263653 If n is the k-th power of a prime then a(n) = k, i.e., a(p^k) = k (p prime, k >= 1): a(A000079(n)) = n, a(A000244(n)) = n, a(A000351(n)) = n, etc.
%C A263653 If n is a squarefree semiprime (A006881) then a(n) - sigma_0(n) = 0, where sigma_0(n) is the number of divisors of n (A000005).
%H A263653 G. C. Greubel, <a href="/A263653/b263653.txt">Table of n, a(n) for n = 2..5000</a>
%H A263653 Ilya Gutkovskiy, <a href="/A263653/a263653.pdf">Bigomega(n)^omega(n)</a>
%H A263653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%H A263653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>
%H A263653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%F A263653 a(n) = A001222(n)^A001221(n).
%F A263653 Sign(a(n)-1) = A066247(n) = A005171(n).
%e A263653 a(30) = 27, because the prime factorization of 30 is 2^1 * 3^1 * 5^1 -> bigomega(30) = 1+1+1, omega(30) = 3 and a(30) = (1+1+1)^3 = 27.
%t A263653 Table[PrimeOmega[n]^PrimeNu[n], {n, 2, 120}]
%o A263653 (PARI) lista(nn) = for(n=2, nn, print1(bigomega(n)^omega(n), ", ")); \\ _Altug Alkan_, Apr 18 2016
%Y A263653 Cf. A000040, A001221, A001222.
%Y A263653 Cf. A046660 (bigomega(n)-omega(n)), A080256 (bigomega(n)+omega(n)), A113901 (bigomega(n)*omega(n)).
%K A263653 nonn
%O A263653 2,3
%A A263653 _Ilya Gutkovskiy_, Apr 17 2016