This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351380 #4 Feb 16 2022 23:25:23
%S A351380 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 A351380 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 A351380 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 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.
%C A351380 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 A351380 Sum_{k>=1} T(n,k) = 2^n.
%F A351380 T(n,2) = A162145(n) for n > 1.
%e A351380 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 A351380 .
%e A351380 A025487(k)| 1 2 4 6 8 12 16 24 30 32 36 48 60 64 72 96 120 ...
%e A351380 ----------+-------------------------------------------------------
%e A351380 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
%e A351380 ----------+-------------------------------------------------------
%e A351380 1 | 1
%e A351380 2 | 0 2
%e A351380 3 | 0 2 1 1
%e A351380 4 | 0 2 1 3 1 1
%e A351380 5 | 0 5 1 3 1 3 1 1 1
%e A351380 6 | 0 7 1 11 0 5 0 3 1 1 1 1 1
%e A351380 7 | 0 13 1 19 1 9 1 2 7 0 1 2 3 1 2 1 1
%e A351380 .
%e A351380 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 A351380 - 0 have prime signature 1 (since all are > 1)
%e A351380 - 5 are primes
%e A351380 - 1 is the square of a prime
%e A351380 - 3 are squarefree semiprimes
%e A351380 etc., as shown below (where p, q, and r represent distinct primes):
%e A351380 .
%e A351380 . prime OEIS
%e A351380 k A025487(k) signature Annnnnn integers in [16, 31] T(5,k)
%e A351380 - ---------- --------- ------- -------------------- ------
%e A351380 1 1 1 - (none) 0
%e A351380 2 2 p A000040 17, 19, 23, 29, 31 5
%e A351380 3 4 p^2 A001248 25 1
%e A351380 4 6 p * q A006881 21, 22, 26 3
%e A351380 5 8 p^3 A030078 27 1
%e A351380 6 12 p^2 * q A054753 18, 20, 28 3
%e A351380 7 16 p^4 A030514 16 1
%e A351380 8 24 p^3 * q A065036 24 1
%e A351380 9 30 p * q * r A007304 30 1
%Y A351380 Cf. A006881, A007304, A025487, A162145.
%K A351380 nonn,tabf
%O A351380 1,3
%A A351380 _Jon E. Schoenfield_, Feb 09 2022