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A263323 The greater of maximal exponent and maximal prime index in the prime factorization of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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

FORMULA

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

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

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

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

Sequence in context: A239514 A304464 A087050 * A263297 A163870 A327664

Adjacent sequences:  A263320 A263321 A263322 * A263324 A263325 A263326

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Oct 14 2015

STATUS

approved

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Last modified April 8 14:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)