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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.
0
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
1,3
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.
FORMULA
Sum_{k>=1} T(n,k) = 2^n.
T(n,2) = A162145(n) for n > 1.
EXAMPLE
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
KEYWORD
nonn,tabf
AUTHOR
Jon E. Schoenfield, Feb 09 2022
STATUS
approved