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A263297
The greater of bigomega(n) and maximal prime index in the prime factorization of n.
27
0, 1, 2, 2, 3, 2, 4, 3, 2, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 3, 4, 5, 9, 4, 3, 6, 3, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 4, 13, 4, 14, 5, 3, 9, 15, 5, 4, 3, 7, 6, 16, 4, 5, 4, 8, 10, 17, 4, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 5, 21, 12, 3, 8, 5, 6, 22, 5
OFFSET
1,3
COMMENTS
Also: minimal m such that n is the product of at most m primes not exceeding prime(m). (Here the primes do not need to be distinct; cf. A263323.)
By convention, a(1)=0, as 1 is the empty product.
Those n with a(n) <= k fill a k-simplex whose 1-sides span from 0 to k.
For a similar construction with distinct primes (hypercube), see A263323.
Each nonnegative integer occurs finitely often; in particular:
- Terms a(n) <= k occur A000984(k) = (2*k)!/(k!)^2 times.
- The term a(n) = 0 occurs exactly once.
- The term a(n) = k > 0 occurs exactly A051924(k) = (3*k-2)*C(k-1) times, where C(k)=A000108(k) are Catalan numbers.
FORMULA
a(n) = max(A001222(n), A061395(n)).
a(n) <= pi(n), with equality when n is 1 or prime.
EXAMPLE
a(6)=2 because 6 is the product of 2 primes (2*3), each not exceeding prime(2)=3.
a(8)=3 because 8 is the product of 3 primes (2*2*2), each not exceeding prime(3)=5.
a(11)=5 because 11 is prime(5).
MAPLE
seq(`if`(n=1, 0, max(pi(max(factorset(n))), bigomega(n))), n=1..80); # Peter Luschny, Oct 15 2015
MATHEMATICA
f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Plus @@ Last /@ FactorInteger@n]; Array[f, 80]
PROG
(PARI) a(n)=if(n<2, return(0)); my(f=factor(n)); max(vecsum(f[, 2]), primepi(f[#f~, 1])) \\ Charles R Greathouse IV, Oct 13 2015
CROSSREFS
Sequence in context: A087050 A341285 A263323 * A163870 A327664 A155043
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Oct 13 2015
STATUS
approved