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

1,6

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

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.]

Prog

(MIT/GNU Scheme)
;; Factorization function ifactor can be found in Antti Karttunen's IntSeq-library, and code for A162319bi given in A162319:
(define (A251725 n) (let ((spf (A020639 n)) (gpf (A006530 n))) (if (= spf gpf) 1 (let outerloop ((k 2)) (let innerloop ((r 1)) (cond ((and (<= r spf) (< gpf (* k r))) k) ((<= r spf) (innerloop (* k r))) (else (outerloop (+ 1 k)))))))))
(define (A251725 n) (if (= 1 n) 1 (let ((fs (uniq (ifactor n)))) (if (= 1 (length fs)) 1 (let outerloop ((base 2)) (let innerloop ((fs fs) (prevlen #f)) (cond ((null? fs) base) ((not prevlen) (innerloop (cdr fs) (A162319bi (car fs) base))) ((= (A162319bi (car fs) base) prevlen) (innerloop (cdr fs) prevlen)) (else (outerloop (+ 1 base))))))))))
(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

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.

Sequence in context: A157118 A156186 A156233 * A292977 A295381 A351420

Adjacent sequences: A251722 A251723 A251724 * A251726 A251727 A251728

Keyword

nonn,base

Author

Antti Karttunen, Dec 16 2014

Status

approved