cp's OEIS Frontend

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

%I A263297 #46 Jan 18 2020 18:22:31
%S A263297 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,
%T A263297 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,
%U A263297 6,6,5,19,7,9,4,20,5,21,12,3,8,5,6,22,5
%N A263297 The greater of bigomega(n) and maximal prime index in the prime factorization of n.
%C A263297 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.)
%C A263297 By convention, a(1)=0, as 1 is the empty product.
%C A263297 Those n with a(n) <= k fill a k-simplex whose 1-sides span from 0 to k.
%C A263297 For a similar construction with distinct primes (hypercube), see A263323.
%C A263297 Each nonnegative integer occurs finitely often; in particular:
%C A263297 - Terms a(n) <= k occur A000984(k) = (2*k)!/(k!)^2 times.
%C A263297 - The term a(n) = 0 occurs exactly once.
%C A263297 - 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.
%H A263297 Antti Karttunen, <a href="/A263297/b263297.txt">Table of n, a(n) for n = 1..20000</a>
%H A263297 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A263297 a(n) = max(A001222(n), A061395(n)).
%F A263297 a(n) <= pi(n), with equality when n is 1 or prime.
%e A263297 a(6)=2 because 6 is the product of 2 primes (2*3), each not exceeding prime(2)=3.
%e A263297 a(8)=3 because 8 is the product of 3 primes (2*2*2), each not exceeding prime(3)=5.
%e A263297 a(11)=5 because 11 is prime(5).
%p A263297 seq(`if`(n=1,0,max(pi(max(factorset(n))),bigomega(n))),n=1..80); # _Peter Luschny_, Oct 15 2015
%t A263297 f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Plus @@ Last /@ FactorInteger@n]; Array[f, 80]
%o A263297 (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
%Y A263297 Cf. A000108, A000984, A001222, A051924, A061395, A263323, A325225, A331296 (ordinal transform), A331297.
%K A263297 nonn
%O A263297 1,3
%A A263297 _Alexei Kourbatov_, Oct 13 2015