A138510 Smallest number b such that in base b the prime factors of the n-th semiprime (A001358) have equal lengths.

1, 2, 1, 6, 8, 3, 3, 12, 1, 14, 12, 18, 2, 20, 14, 24, 1, 18, 4, 20, 30, 32, 4, 24, 38, 4, 42, 5, 44, 30, 4, 32, 48, 5, 54, 38, 5, 60, 5, 1, 62, 42, 44, 5, 68, 48, 72, 2, 30, 74, 32, 80, 54, 5, 84, 1, 60, 90, 62, 38, 3, 98, 68, 102, 6, 42, 104, 3, 72, 108, 44, 6, 110, 74, 3, 114, 48, 80

Offset

1,2

Comments

a(n) = 1 iff A001358(n) is the square of a prime (A001248);

a(A174956(A138511(n))) = A084127(A174956(A138511(n))) + 1.

Equally, 1 if A001358(n) = p^2, otherwise, if A001358(n) = p*q (p, q primes, p < q), then a(n) = A252375(n) = the least r such that r^k <= p < q < r^(k+1), for some k >= 0. - Antti Karttunen, Dec 16 2014

a(A174956(A085721(n))) <= 2. - Reinhard Zumkeller, Dec 19 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = A251725(A001358(n)). - Antti Karttunen, Dec 16 2014

Example

For n=31, the n-th semiprime is A001358(31) = 91 = 7*13;
     7 =  111_2 =  21_3 = 13_4
and 13 = 1101_2 = 111_3 = 31_4, so a(31) = 4. [corrected by Jon E. Schoenfield, Sep 23 2018]
.
Illustration of initial terms, n <= 25:
.   n | A001358(n) =  p * q |  b = a(n) | p and q in base b
. ----+---------------------+-----------+-------------------
.   1 |       4       2   2 |      1    |     [1]        [1]
.   2 |       6       2   3 |      2    |   [1,0]      [1,1]
.   3 |       9       3   3 |      1    | [1,1,1]    [1,1,1]
.   4 |  **  10       2   5 |      6    |     [2]        [5]
.   5 |  **  14       2   7 |      8    |     [2]        [7]
.   6 |      15       3   5 |      3    |   [1,0]      [1,2]
.   7 |      21       3   7 |      3    |   [1,0]      [2,1]
.   8 |  **  22       2  11 |     12    |     [2]       [11]
.   9 |      25       5   5 |      1    |   [1]^5      [1]^5
.  10 |  **  26       2  13 |     14    |     [2]       [13]
.  11 |  **  33       3  11 |     12    |     [3]       [11]
.  12 |  **  34       2  17 |     18    |     [2]       [17]
.  13 |      35       5   7 |      2    | [1,0,1]    [1,1,1]
.  14 |  **  38       2  19 |     20    |     [2]       [19]
.  15 |  **  39       3  13 |     14    |     [3]       [13]
.  16 |  **  46       2  23 |     24    |     [2]       [23]
.  17 |      49       7   7 |      1    |   [1]^7      [1]^7
.  18 |  **  51       3  17 |     18    |     [3]       [17]
.  19 |      55       5  11 |      4    |   [1,1]      [2,3]
.  20 |  **  57       3  19 |     20    |     [3]       [19]
.  21 |  **  58       2  29 |     30    |     [2]       [29]
.  22 |  **  62       2  31 |     32    |     [2]       [31]
.  23 |      65       5  13 |      4    |   [1,1]      [3,1]
.  24 |  **  69       3  23 |     24    |     [3]       [23]
.  25 |  **  74       2  37 |     38    |     [2]       [37]
where p = A084126(n) and q = A084127(n),
semiprimes marked with ** indicate terms of A138511, i.e. b = q + 1.

Prog

(Haskell)
import Data.List (genericIndex, unfoldr); import Data.Tuple (swap)
import Data.Maybe (mapMaybe)
a138510 n = genericIndex a138510_list (n - 1)
a138510_list = mapMaybe f [1..] where
  f x | a010051' q == 0 = Nothing
      | q == p          = Just 1
      | otherwise       = Just $
        head [b | b <- [2..], length (d b p) == length (d b q)]
      where q = div x p; p = a020639 x
  d b = unfoldr (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b)
-- Reinhard Zumkeller, Dec 16 2014
(Scheme) (define (A138510 n) (A251725 (A001358 n))) ;; Antti Karttunen, Dec 16 2014

Crossrefs

Cf. A078972, A085721.

Cf. A138511, A251728, A252375, A084127, A020639, A162319, A174956, A001358.

Sequence in context: A363747 A220884 A217877 * A026215 A026220 A138750

Adjacent sequences: A138507 A138508 A138509 * A138511 A138512 A138513

Keyword

nonn,base

Author

Reinhard Zumkeller, Mar 21 2008

Extensions

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014

Status

approved