A283866 Multiplicities of prime factors of A243103(n).

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

Offset: 1

Author

Michael De Vlieger, Mar 17 2017

Keywords

• nonn
• tabf
• easy

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).

Examples

			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
...
		

Links

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

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).

Programs

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 *)