A263297 The greater of bigomega(n) and maximal prime index in the prime factorization of n.

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

Author

Alexei Kourbatov, Oct 13 2015

Keywords

• nonn

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.

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000108, A000984, A001222, A051924, A061395, A263323, A325225, A331296 (ordinal transform), A331297.

Programs

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]

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

Formula

a(n) = max(A001222(n), A061395(n)).

a(n) <= pi(n), with equality when n is 1 or prime.