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A249809 Irregular table read by rows: T(n, k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1 .. n. (T(1,1) = 0, and for each n > 1, k = 1 .. A000720(n)).

%I A249809 #48 Apr 10 2024 09:49:39
%S A249809 0,1,1,1,2,1,2,1,1,3,1,1,3,1,1,1,4,1,1,1,4,2,1,1,5,2,1,1,5,2,1,1,1,6,
%T A249809 2,1,1,1,6,2,1,1,1,1,7,2,1,1,1,1,7,3,1,1,1,1,8,3,1,1,1,1,8,3,1,1,1,1,
%U A249809 1,9,3,1,1,1,1,1,9,3,1,1,1,1,1,1,10,3,1,1,1,1,1,1,10,4,1,1,1,1,1,1,11,4,1,1,1,1,1,1,11,4,1,1,1,1,1,1,1
%N A249809 Irregular table read by rows: T(n, k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1 .. n. (T(1,1) = 0, and for each n > 1, k = 1 .. A000720(n)).
%C A249809 After the first row {0}, consists of rows of triangular table A249808 with trailing zeros removed.
%H A249809 Antti Karttunen, <a href="/A249809/b249809.txt">Table of n, a(n) for n = 1..10016; the first 294 rows of the table, flattened</a>
%H A249809 Orges Leka, <a href="/A249809/a249809.png">Fractal of prime factorization of integers, lexicographically sorted</a>.
%H A249809 Orges Leka, <a href="https://mathoverflow.net/questions/458845/">How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?</a>
%F A249809 a(n) = A249808(A249728(n), A249727(n)).
%F A249809 For n > 1, A078898(n) = T(n, A055396(n)).
%e A249809 Table begins:
%e A249809        k=1  2  3  4  5  6  7
%e A249809   n=1:   0;
%e A249809   n=2:   1;
%e A249809   n=3:   1, 1;
%e A249809   n=4:   2, 1;
%e A249809   n=5:   2, 1, 1;
%e A249809   n=6:   3, 1, 1;
%e A249809   n=7:   3, 1, 1, 1;
%e A249809   n=8:   4, 1, 1, 1;
%e A249809   n=9:   4, 2, 1, 1;
%e A249809   n=10:  5, 2, 1, 1;
%e A249809   n=11:  5, 2, 1, 1, 1;
%e A249809   n=12:  6, 2, 1, 1, 1;
%e A249809   n=13:  6, 2, 1, 1, 1, 1;
%e A249809   n=14:  7, 2, 1, 1, 1, 1;
%e A249809   n=15:  7, 3, 1, 1, 1, 1;
%e A249809   n=16:  8, 3, 1, 1, 1, 1;
%e A249809   n=17:  8, 3, 1, 1, 1, 1, 1;
%e A249809   ...
%o A249809 (Scheme) (define (A249809 n) (A249808bi (A249728 n) (A249727 n))) ;; Code for A249808bi given in A249808.
%Y A249809 A004526 gives the left edge, A001477 the row sums.
%Y A249809 Cf. A000040, A000720, A020639, A249808, A249727, A249728, A055396, A078898.
%K A249809 nonn,tabf
%O A249809 1,5
%A A249809 _Antti Karttunen_, Nov 06 2014