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A263653
a(n) = bigomega(n)^omega(n).
3
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
2,3
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).
LINKS
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
FORMULA
a(n) = A001222(n)^A001221(n).
Sign(a(n)-1) = A066247(n) = A005171(n).
EXAMPLE
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.
MATHEMATICA
Table[PrimeOmega[n]^PrimeNu[n], {n, 2, 120}]
PROG
(PARI) lista(nn) = for(n=2, nn, print1(bigomega(n)^omega(n), ", ")); \\ Altug Alkan, Apr 18 2016
CROSSREFS
Cf. A046660 (bigomega(n)-omega(n)), A080256 (bigomega(n)+omega(n)), A113901 (bigomega(n)*omega(n)).
Sequence in context: A062799 A063647 A353379 * A330328 A269427 A349391
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 17 2016
STATUS
approved