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%I #4 Feb 16 2022 23:25:23
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%F Sum_{k>=1} T(n,k) = 2^n.
%F T(n,2) = A162145(n) for n > 1.
%e 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.
%e .
%e A025487(k)| 1 2 4 6 8 12 16 24 30 32 36 48 60 64 72 96 120 ...
%e ----------+-------------------------------------------------------
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
%e ----------+-------------------------------------------------------
%e 1 | 1
%e 2 | 0 2
%e 3 | 0 2 1 1
%e 4 | 0 2 1 3 1 1
%e 5 | 0 5 1 3 1 3 1 1 1
%e 6 | 0 7 1 11 0 5 0 3 1 1 1 1 1
%e 7 | 0 13 1 19 1 9 1 2 7 0 1 2 3 1 2 1 1
%e .
%e 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]:
%e - 0 have prime signature 1 (since all are > 1)
%e - 5 are primes
%e - 1 is the square of a prime
%e - 3 are squarefree semiprimes
%e etc., as shown below (where p, q, and r represent distinct primes):
%e .
%e . prime OEIS
%e k A025487(k) signature Annnnnn integers in [16, 31] T(5,k)
%e - ---------- --------- ------- -------------------- ------
%e 1 1 1 - (none) 0
%e 2 2 p A000040 17, 19, 23, 29, 31 5
%e 3 4 p^2 A001248 25 1
%e 4 6 p * q A006881 21, 22, 26 3
%e 5 8 p^3 A030078 27 1
%e 6 12 p^2 * q A054753 18, 20, 28 3
%e 7 16 p^4 A030514 16 1
%e 8 24 p^3 * q A065036 24 1
%e 9 30 p * q * r A007304 30 1
%Y Cf. A006881, A007304, A025487, A162145.
%K nonn,tabf
%O 1,3
%A _Jon E. Schoenfield_, Feb 09 2022