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A283866
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Multiplicities of prime factors of A243103(n).
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2
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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
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OFFSET
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1,4
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COMMENTS
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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.
a(1) = 0; a(p) = 1 for prime p. For prime powers p^e with e>=0, a(p^e) = A000217(e).
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LINKS
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EXAMPLE
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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.
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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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