|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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.
|
|
|
LINKS
|
|
|
|
FORMULA
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
