A351380 Table read by rows: T(n,k) is the number of integers in the interval [2^(n-1), 2^n - 1] that have the k-th least prime signature.

1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 3, 1, 1, 0, 5, 1, 3, 1, 3, 1, 1, 1, 0, 7, 1, 11, 0, 5, 0, 3, 1, 1, 1, 1, 1, 0, 13, 1, 19, 1, 9, 1, 2, 7, 0, 1, 2, 3, 1, 2, 1, 1, 0, 23, 1, 39, 0, 14, 0, 8, 16, 1, 2, 3, 9, 0, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 43, 2, 73, 1, 27, 0, 11, 37, 0, 2, 6, 20, 0, 2, 3, 8, 0, 2, 4, 2, 4, 0, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Author

Jon E. Schoenfield, Feb 09 2022

Keywords

• nonn
• tabf

Comments

In rows n = 4 and n = 6..19, T(n,4) is the largest term in the row, i.e., squarefree semiprimes (A006881) outnumber the integers of each of the other prime signatures, but T(20,4) = 106408 < 109245 = T(20,9): among 20-bit numbers, sphenic numbers (A007304) (i.e., products of three distinct primes) are more numerous than squarefree semiprimes.

Examples

			The first 7 rows are shown in the body of the table below. Across the top of the table are the terms of A025487, whose k-th term is the smallest integer having the k-th prime signature.
.
  A025487(k)| 1  2  4  6  8 12 16 24 30 32 36 48 60 64 72 96 120 ...
  ----------+-------------------------------------------------------
         n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17  ...
  ----------+-------------------------------------------------------
         1  | 1
         2  | 0  2
         3  | 0  2  1  1
         4  | 0  2  1  3  1  1
         5  | 0  5  1  3  1  3  1  1  1
         6  | 0  7  1 11  0  5  0  3  1  1  1  1  1
         7  | 0 13  1 19  1  9  1  2  7  0  1  2  3  1  2  1  1
.
E.g., the 9 terms in row n=5 are 0, 5, 1, 3, 1, 3, 1, 1, 1 because, of the 16 integers in the interval [2^(5-1), 2^5 - 1] = [16, 31]:
  - 0 have prime signature 1 (since all are > 1)
  - 5 are primes
  - 1 is the square of a prime
  - 3 are squarefree semiprimes
etc., as shown below (where p, q, and r represent distinct primes):
.
.                  prime     OEIS
  k  A025487(k)  signature  Annnnnn  integers in [16, 31]  T(5,k)
  -  ----------  ---------  -------  --------------------  ------
  1       1          1         -            (none)            0
  2       2          p      A000040   17, 19, 23, 29, 31      5
  3       4         p^2     A001248           25              1
  4       6        p * q    A006881       21, 22, 26          3
  5       8         p^3     A030078           27              1
  6      12       p^2 * q   A054753       18, 20, 28          3
  7      16         p^4     A030514           16              1
  8      24       p^3 * q   A065036           24              1
  9      30      p * q * r  A007304           30              1
		

Crossrefs

Cf. A006881, A007304, A025487, A162145.

Formula

Sum_{k>=1} T(n,k) = 2^n.

T(n,2) = A162145(n) for n > 1.