A251725 Smallest number b such that in base-b representation the prime factors of n have equal lengths.

1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 8, 3, 1, 1, 2, 1, 6, 3, 12, 1, 2, 1, 14, 1, 8, 1, 6, 1, 1, 12, 18, 2, 2, 1, 20, 14, 6, 1, 8, 1, 12, 3, 24, 1, 2, 1, 6, 18, 14, 1, 2, 4, 8, 20, 30, 1, 6, 1, 32, 3, 1, 4, 12, 1, 18, 24, 8, 1, 2, 1, 38, 3, 20, 4, 14, 1, 6, 1, 42, 1, 8, 5, 44, 30, 12, 1, 6, 4, 24, 32, 48, 5, 2, 1, 8, 12, 6

Offset: 1

Author

Antti Karttunen, Dec 16 2014

Keywords

• nonn
• base

Comments

The "base-1" is here same as "unary base", where n is represented with digit "1" replicated n times. Thus if and only if n is in A000961 (is a power of prime), a(n) = 1. See A252375 for a more consistent treatment of those cases.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A252375 (variant).

Cf. A251726 (those n > 1 for which a(n) <= A006530(n)).

Cf. A251727 (those n for which a(n) = A006530(n)+1).

Cf. A000961 (positions of ones).

Cf. A001358, A006530, A020639, A066048, A138510, A162319.

Cf. A027748.

Programs

Haskell

import Data.List (unfoldr); import Data.Tuple (swap)
a251725 1 = 1
a251725 n = if length ps == 1 then 1 else head $ filter f [2..]  where
  f b = all (== len) lbs where len:lbs = map (length . d b) ps
  ps = a027748_row n
  d b = unfoldr (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b)
-- Reinhard Zumkeller, Dec 17 2014

Formula

Other identities. For all n >= 1:

A138510(n) = a(A001358(n)).

a(n) = a(A066048(n)). [The result depends only on the smallest and the largest prime factor of n.]