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Multiplicities of prime factors of A243103(n).
2

%I #12 Mar 18 2017 11:33:54

%S 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 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 1,28,15,5,1,18,3,9,5,16,2,1,28,14,3,22,8

%N Multiplicities of prime factors of A243103(n).

%C Irregular triangle T(n,m) of multiplicities of the product of the numbers 1 <= k <= n | n^e with e >= 0.

%C Count of instances of primes p|n among the prime factors of all numbers 1 <= k <= n.

%C A243103(n) = Product of row n of A162306; prime divisors of A243103(n) = prime divisors of n = A027748(n).

%C a(1) = 0; a(p) = 1 for prime p. For prime powers p^e with e>=0, a(p^e) = A000217(e).

%H Michael De Vlieger, <a href="/A283866/b283866.txt">Table of n, a(n) for n = 1..11739</a> (rows 1 <= n <= 5000)

%e 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 1 = 1

%e 2 = 2^1

%e 3 = 3^1

%e 4 = 2^2

%e 6 = 2^1 * 3^1

%e 8 = 2^3

%e 9 = 3^2

%e 12 = 2^2 * 3^1

%e Adding the multiplicities of the factors of 12 for each of these gives us 9,5.

%e Row 42 = 28,15,5 since A243103(42) = 64736452901018271744 = 2^28 * 3^15 * 7^5.

%e Relationship of first 12 rows of a(n) with A027748(n) and A243103(n):

%e n A027748(n) a(n) A243103(n)

%e 1 1 0 1 = 1^0

%e 2 2 1 2 = 2^1

%e 3 3 1 3 = 3^1

%e 4 2 3 8 = 2^3

%e 5 5 1 5 = 5^1

%e 6 2,3 4,2 144 = 2^4 * 3^2

%e 7 7 1 7 = 7^1

%e 8 2 6 64 = 2^6

%e 9 3 3 27 = 3^3

%e 10 2,5 7,2 3200 = 2^7 * 5*2

%e 11 11 1 11 = 11^1

%e 12 2,3 9,5 124416 = 2^9 * 3^5

%e ...

%e T(n,m) for n = primorial p_x# = A002110(x), with horizontal axis the multiplicity pertaining to prime(m):

%e x 2 3 5 7 11 13 17 19

%e 1 1

%e 2 4 2

%e 3 21 13 7

%e 4 118 63 36 26

%e 5 625 351 200 147 101

%e 6 2982 1694 1003 753 537 477

%e 7 14131 8128 4905 3733 2693 2404 2025

%e 8 64332 37274 22763 17448 12744 11450 9698 9078

%e ...

%t 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 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 nonn,tabf,easy

%O 1,4

%A _Michael De Vlieger_, Mar 17 2017