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A283866 Multiplicities of prime factors of A243103(n). 2
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, 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, 1, 28, 15, 5, 1, 18, 3, 9, 5, 16, 2, 1, 28, 14, 3, 22, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

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

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

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

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..11739 (rows 1 <= n <= 5000)

EXAMPLE

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:

   1 = 1

   2 = 2^1

   3 =       3^1

   4 = 2^2

   6 = 2^1 * 3^1

   8 = 2^3

   9 =       3^2

  12 = 2^2 * 3^1

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

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

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

   n   A027748(n) a(n) A243103(n)

   1    1         0          1 = 1^0

   2    2         1          2 = 2^1

   3    3         1          3 = 3^1

   4    2         3          8 = 2^3

   5    5         1          5 = 5^1

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

   7    7         1          7 = 7^1

   8    2         6         64 = 2^6

   9    3         3         27 = 3^3

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

  11   11         1         11 = 11^1

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

  ...

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

  x      2       3       5       7      11      13      17      19

  1      1

  2      4       2

  3     21      13       7

  4    118      63      36      26

  5    625     351     200     147     101

  6   2982    1694    1003     753     537     477

  7  14131    8128    4905    3733    2693    2404    2025

  8  64332   37274   22763   17448   12744   11450    9698    9078

...

MATHEMATICA

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 *)

CROSSREFS

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).

Sequence in context: A131033 A135821 A289205 * A134545 A174907 A237016

Adjacent sequences:  A283863 A283864 A283865 * A283867 A283868 A283869

KEYWORD

nonn,tabf,easy

AUTHOR

Michael De Vlieger, Mar 17 2017

STATUS

approved

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Last modified June 4 11:32 EDT 2020. Contains 334825 sequences. (Running on oeis4.)