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%I A283866 #12 Mar 18 2017 11:33:54
%S A283866 0,1,1,3,1,4,2,1,6,3,7,2,1,9,5,1,7,2,4,2,10,1,14,7,1,13,3,4,2,11,2,1,
%T A283866 17,8,3,11,2,6,13,3,1,21,13,7,1,15,7,2,16,2,4,2,24,13,1,16,2,7,2,21,6,
%U A283866 1,28,15,5,1,18,3,9,5,16,2,1,28,14,3,22,8
%N A283866 Multiplicities of prime factors of A243103(n).
%C A283866 Irregular triangle T(n,m) of multiplicities of the product of the numbers 1 <= k <= n | n^e with e >= 0.
%C A283866 Count of instances of primes p|n among the prime factors of all numbers 1 <= k <= n.
%C A283866 A243103(n) = Product of row n of A162306; prime divisors of A243103(n) = prime divisors of n = A027748(n).
%C A283866 a(1) = 0; a(p) = 1 for prime p. For prime powers p^e with e>=0, a(p^e) = A000217(e).
%H A283866 Michael De Vlieger, <a href="/A283866/b283866.txt">Table of n, a(n) for n = 1..11739</a> (rows 1 <= n <= 5000)
%e A283866 Row 12 = 9,5 because the numbers 1 <= k <= 12 | 12^e with e>=0, {1,2,3,4,6,8,9,12} and these have the prime decompositions:
%e A283866    1 = 1
%e A283866    2 = 2^1
%e A283866    3 =       3^1
%e A283866    4 = 2^2
%e A283866    6 = 2^1 * 3^1
%e A283866    8 = 2^3
%e A283866    9 =       3^2
%e A283866   12 = 2^2 * 3^1
%e A283866 Adding the multiplicities of the factors of 12 for each of these gives us 9,5.
%e A283866 Row 42 = 28,15,5 since A243103(42) = 64736452901018271744 = 2^28 * 3^15 * 7^5.
%e A283866 Relationship of first 12 rows of a(n) with A027748(n) and A243103(n):
%e A283866    n   A027748(n) a(n) A243103(n)
%e A283866    1    1         0          1 = 1^0
%e A283866    2    2         1          2 = 2^1
%e A283866    3    3         1          3 = 3^1
%e A283866    4    2         3          8 = 2^3
%e A283866    5    5         1          5 = 5^1
%e A283866    6    2,3       4,2      144 = 2^4 * 3^2
%e A283866    7    7         1          7 = 7^1
%e A283866    8    2         6         64 = 2^6
%e A283866    9    3         3         27 = 3^3
%e A283866   10    2,5       7,2     3200 = 2^7 * 5*2
%e A283866   11   11         1         11 = 11^1
%e A283866   12    2,3       9,5   124416 = 2^9 * 3^5
%e A283866   ...
%e A283866 T(n,m) for n = primorial p_x# = A002110(x), with horizontal axis the multiplicity pertaining to prime(m):
%e A283866   x      2       3       5       7      11      13      17      19
%e A283866   1      1
%e A283866   2      4       2
%e A283866   3     21      13       7
%e A283866   4    118      63      36      26
%e A283866   5    625     351     200     147     101
%e A283866   6   2982    1694    1003     753     537     477
%e A283866   7  14131    8128    4905    3733    2693    2404    2025
%e A283866   8  64332   37274   22763   17448   12744   11450    9698    9078
%e A283866 ...
%t A283866 Table[With[{m = Floor@ Log2@ n}, Values@ Merge[Association /@ Map[#1 -> #2 & @@ # &, FactorInteger@ Rest@ Select[Range@ n, PowerMod[n, m, #] == 0 &], {2}], Total]] /. {} -> {0}, {n, 50}] // Flatten (* _Michael De Vlieger_, Mar 17 2017, Version 10 *)
%Y A283866 Cf. A001221 (row lengths), A027748, A243103, A010846 (number of 1 <= m <= n | n^e), A162306 (list of 1 <= m <= n | n^e), A124010 (multiplicities of primes in n).
%K A283866 nonn,tabf,easy
%O A283866 1,4
%A A283866 _Michael De Vlieger_, Mar 17 2017

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