A162319 -id:A162319 - cp's OEIS frontend

A138511 Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.

10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 291, 295, 298, 302, 303, 305

Offset: 1

Reinhard Zumkeller, Mar 21 2008

From Antti Karttunen, Dec 17 2014, further edited Jan 01 & 04 2014: (Start)
Semiprimes p*q, p < q, such that the smallest r for which r^k <= p and q < r^(k+1) [for some k >= 0] is q+1, and thus k = 0. In other words, semiprimes whose both prime factors do not fit (simultaneously) between any two consecutive powers of any natural number r less than or equal to the larger prime factor. This condition forces the larger prime factor q to be greater than the square of the smaller prime factor because otherwise the opposite condition given in A251728 would hold.
Assuming that A054272(n), the number of primes in interval [prime(n), prime(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true), these are also "unsettled" semiprimes that occur in a square array A083221 constructed from the sieve of Eratosthenes, "above the line A251719", meaning that if and only if row < A251719(col) then a semiprime occurring at A083221(row, col) is in this sequence, and conversely, all the semiprimes that occur at any position A083221(row, col) where row >= A251719(col) are in the complementary sequence A251728.
(End)
Semiprimes p*q, p < q, such that b = q+1 is the minimal base with the property that p and q have equal length representations in base b. This was the original definition, which is based primarily on A138510: A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

			See A138510.

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

Cf. A138510.
Complement of A251728 in A001358.
Subsequence of A088381.
An intersection of A001358 (semiprimes) and A251727.
Also an intersection of A001358 and A253569, from the latter which this sequence differs for the first time at n=60, where A253569(60) = 290, while here a(60) = 291.
Also an intersection A001358 and A245729.
Cf. also A006530, A020639, A054272, A083221, A083140, A084127, A138510, A162319, A174956, A251719.

a138511 n = a138511_list !! (n-1)
a138511_list = filter f [1..] where
                      f x = p ^ 2 < q && a010051' q == 1
                            where q = div x p; p = a020639 x
-- Reinhard Zumkeller, Jan 06 2015

Select[Range[305],PrimeOmega[#]==2&&Divisors[#][[-2]]>Divisors[#][[-3]]^2&&SquareFreeQ[#]&] (* James C. McMahon, Jun 07 2025 *)

isok(s) = my(f=factor(s)); (bigomega(f) == 2) && (#f~ == 2) && (f[1,1]^2 < f[2, 1]); \\ Michel Marcus, Sep 15 2020

;; Scheme with Antti Karttunen's IntSeq-library.
(define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (= (A252375 n) (+ 1 (A006530 n)))))))
(define A138511 (COMPOSE A001358 (MATCHING-POS 1 1 (lambda (n) (= (A138510 n) (+ 1 (A006530 (A001358 n))))))))
;; Antti Karttunen, Dec 16-17 2014

;; Scheme with Antti Karttunen's IntSeq-library.
(define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (> (A006530 n) (A000290 (A020639 n))))))) ;; Based on the new alternative definition - Antti Karttunen, Jan 01 2015

Other identities. For all n >= 1 it holds that:

A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014
New definition by Antti Karttunen, Jan 01 2015; old definition moved to comment.
More terms from Antti Karttunen, Jan 09 2015

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

Original entry on oeis.org

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

Reinhard Zumkeller, Mar 21 2008

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

			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.

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

Cf. A078972, A085721.

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

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

(define (A138510 n) (A251725 (A001358 n))) ;; Antti Karttunen, Dec 16 2014

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

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014

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

Original entry on oeis.org

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

Antti Karttunen, Dec 16 2014

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.

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

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.

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

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.]

A162320 Array read by antidiagonals: a(n,m) = the number of digits of m when written in base n. The top row is the number of digits for each m in base 2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2

Offset: 1

Leroy Quet, Jul 01 2009

A162319 is the same array with the lengths of base 1 numbers in the top row.

			From _Michael De Vlieger_, Jan 02 2015: (Start)
Array read by antidiagonals begins:
  1;
  1, 2;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 1, 2, 3;
  1, 1, 1, 2, 2, 3;
  1, 1, 1, 1, 2, 2, 3;
  1, 1, 1, 1, 2, 2, 2, 4;
  1, 1, 1, 1, 1, 2, 2, 2, 4;
  ...
Array adjusted such that the rows represent base n and the columns m:
                         m
           1  2  3  4  5  6  7  8  9  10
           ------------------------------
  base 2:  1, 2, 2, 3, 3, 3, 3, 4, 4, (4);
  base 3:  1, 1, 2, 2, 2, 2, 2, 2, (3, 3);
  base 4:  1, 1, 1, 2, 2, 2, 2, (2, 2, 2);
  base 5:  1, 1, 1, 1, 2, 2, (2, 2, 2, 2);
  base 6:  1, 1, 1, 1, 1, (2, 2, 2, 2, 2);
  base 7:  1, 1, 1, 1, (1, 1, 2, 2, 2, 2);
  base 8:  1, 1, 1, (1, 1, 1, 1, 2, 2, 2);
  base 9:  1, 1, (1, 1, 1, 1, 1, 1, 2, 2);
  base 10: 1, (1, 1, 1, 1, 1, 1, 1, 1, 1);
  ...
For n = 12, a(12) is found in the second position in row 5 in the array read by antidiagonals. This equates to m = 2, base n = 5. The number m = 2 in base n = 5 requires 1 digit, thus a(12) = 1.
For n = 20, a(20) is found in the fifth position in row 6 in the array read by antidiagonals. This equates to m = 5, base n = 3. The number m = 5 in base n = 3 requires 2 digits, thus a(20) = 2. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10296 (Covers bases n = 1..144)

Cf. A162319.

a162320[n_] := Block[{t = {}, i, j}, For[i = 1, i <= n, i++, For[j = i, j > 1, j--, AppendTo[t, Floor@Log[j, i - j + 1] + 1]]]; t]; a162320[14] (* Michael De Vlieger, Jan 02 2015 *)

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A138511 Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.

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

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

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A162320 Array read by antidiagonals: a(n,m) = the number of digits of m when written in base n. The top row is the number of digits for each m in base 2.

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