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

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, 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, 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

Offset: 2

Author

Ilya Gutkovskiy, Apr 17 2016

Keywords

• nonn

Comments

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).

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.

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).

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5000
• Ilya Gutkovskiy, Bigomega(n)^omega(n)
• Eric Weisstein's World of Mathematics, Distinct Prime Factors
• Eric Weisstein's World of Mathematics, Prime Factor
• Eric Weisstein's World of Mathematics, Prime Power

Crossrefs

Cf. A000040, A001221, A001222.

Cf. A046660 (bigomega(n)-omega(n)), A080256 (bigomega(n)+omega(n)), A113901 (bigomega(n)*omega(n)).

Programs

Mathematica

Table[PrimeOmega[n]^PrimeNu[n], {n, 2, 120}]

PARI

lista(nn) = for(n=2, nn, print1(bigomega(n)^omega(n), ", ")); \\ Altug Alkan, Apr 18 2016

Formula

a(n) = A001222(n)^A001221(n).

Sign(a(n)-1) = A066247(n) = A005171(n).