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

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, 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, 6, 6, 5, 19, 7, 9, 4, 20, 3, 21, 12, 3, 8, 5, 6, 22, 4

Offset: 1

Author

Alexei Kourbatov, Oct 14 2015

Keywords

• nonn

Comments

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

Also: minimal m such that n is the product of at most m distinct primes not exceeding prime(m), with multiplicity at most m.

By convention, a(1)=0, as 1 is the empty product.

Those n with a(n) <= k fill a k-hypercube whose 1-sides span from 0 to k.

A263297 is a similar construction, with a k-simplex instead of a hypercube.

Each nonnegative integer occurs finitely often; in particular:

- Terms a(n) <= k occur A000169(k+1) = (k+1)^k times.

- The term a(n) = 0 occurs exactly once.

- The term a(n) = k > 0 occurs exactly A178922(k) = (k+1)^k - k^(k-1) times.

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000169, A001221, A002110, A051903, A061395, A178922, A263297.

Programs

Mathematica

f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Max @@ Last /@ FactorInteger@n]; Array[f, 80]

PARI

a(n) = if (n==1, 0, my(f = factor(n)); max(vecmax(f[,2]), primepi(f[#f~,1]))); \\ Michel Marcus, Oct 15 2015

Formula

a(n) = max(A051903(n), A061395(n)).

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