OFFSET
1,3
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
EXAMPLE
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.
MATHEMATICA
f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Max @@ Last /@ FactorInteger@n]; Array[f, 80]
PROG
(PARI) a(n) = if (n==1, 0, my(f = factor(n)); max(vecmax(f[, 2]), primepi(f[#f~, 1]))); \\ Michel Marcus, Oct 15 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Oct 14 2015
STATUS
approved