cp's OEIS Frontend

A263323 The greater of maximal exponent and maximal prime index in the prime factorization of n.

%I A263323 #37 May 18 2017 02:41:13
%S A263323 0,1,2,2,3,2,4,3,2,3,5,2,6,4,3,4,7,2,8,3,4,5,9,3,3,6,3,4,10,3,11,5,5,
%T A263323 7,4,2,12,8,6,3,13,4,14,5,3,9,15,4,4,3,7,6,16,3,5,4,8,10,17,3,18,11,4,
%U A263323 6,6,5,19,7,9,4,20,3,21,12,3,8,5,6,22,4
%N A263323 The greater of maximal exponent and maximal prime index in the prime factorization of n.
%C A263323 Also: minimal m such that n divides (prime(m)#)^m. Here prime(m)# denotes the primorial A002110(m), i.e., the product of all primes from 2 to prime(m). - _Charles R Greathouse IV_, Oct 15 2015
%C A263323 Also: minimal m such that n is the product of at most m distinct primes not exceeding prime(m), with multiplicity at most m.
%C A263323 By convention, a(1)=0, as 1 is the empty product.
%C A263323 Those n with a(n) <= k fill a k-hypercube whose 1-sides span from 0 to k.
%C A263323 A263297 is a similar construction, with a k-simplex instead of a hypercube.
%C A263323 Each nonnegative integer occurs finitely often; in particular:
%C A263323 - Terms a(n) <= k occur A000169(k+1) = (k+1)^k times.
%C A263323 - The term a(n) = 0 occurs exactly once.
%C A263323 - The term a(n) = k > 0 occurs exactly A178922(k) = (k+1)^k - k^(k-1) times.
%H A263323 G. C. Greubel, <a href="/A263323/b263323.txt">Table of n, a(n) for n = 1..5000</a>
%F A263323 a(n) = max(A051903(n), A061395(n)).
%F A263323 a(n) <= pi(n), with equality if n=1 or prime.
%e A263323 a(36)=2 because 36 is the product of 2 distinct primes (2*2*3*3), each not exceeding prime(2)=3, with multiplicity not exceeding 2.
%t A263323 f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Max @@ Last /@ FactorInteger@n]; Array[f, 80]
%o A263323 (PARI) a(n) = if (n==1, 0, my(f = factor(n)); max(vecmax(f[,2]), primepi(f[#f~,1]))); \\ _Michel Marcus_, Oct 15 2015
%Y A263323 Cf. A000169, A001221, A002110, A051903, A061395, A178922, A263297.
%K A263323 nonn
%O A263323 1,3
%A A263323 _Alexei Kourbatov_, Oct 14 2015