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

1,3

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization

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

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

Sequence in context: A087050 A341285 A263323 * A163870 A327664 A155043

Adjacent sequences: A263294 A263295 A263296 * A263298 A263299 A263300

Keyword

nonn

Author

Alexei Kourbatov, Oct 13 2015

Status

approved