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A363596
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a(n) = (Product_{k=1..pi(n+1)} prime(k)^floor(n/(prime(k)-1) ) )/(n+1)!.
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1
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1, 1, 2, 1, 6, 2, 12, 3, 10, 2, 12, 2, 420, 60, 24, 3, 90, 10, 420, 42, 660, 60, 360, 30, 3276, 252, 56, 4, 120, 8, 3696, 231, 3570, 210, 36, 2, 103740, 5460, 840, 42, 13860, 660, 27720, 1260, 19320, 840, 5040, 210, 198900, 7956, 10296, 396, 11880, 440, 6384, 228
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OFFSET
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0,3
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COMMENTS
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Motivated by Proposition 3.2, p. 10 of the Bedhouche-Farhi paper.
Observations regarding prime power decomposition of terms in a(0..20737):
For n > 300, most terms are in A361098 but not in A286708. A303606 is a subset of A286708, which is a subset of A361098, which in turn is a subset of A126706, numbers that are neither prime powers nor squarefree.
a(34) = 36 is the only term in A286708 (more specifically, in A303606).
a(35) = 2 is the last prime term.
a(29) = 8 is the only composite prime power.
a(190) = 221760 is the last term in A002182, but a(61) = a(102) = 720720 is the largest.
a(191) = 2310 is the last primorial term.
a(1055) = 2207550414530882190 is the last squarefree term. If there are further squarefree terms a(n), n is likely to belong to -1 (mod 24).
a(7055) = 1733187515208453605856007490304335826298500960 is the last term that is not in A361098. a(n) not in A361098 is likely to belong to -1 (mod 24).
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LINKS
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Michael De Vlieger, Plot p(k)^e(k) | a(n) at (x, y) = (n, k), n = 0..2^11, with a color function representing e(k), where black = 1, red = 2, and the largest exponent in the dataset shown in magenta. The bar at bottom shows the number 1 in black, primes in red, composite prime powers in gold, squarefree terms in green, and terms that are neither squarefree nor prime powers in blue.
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FORMULA
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EXAMPLE
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The table below relates b(n) = A091137(n) to a(n), with (n+1)!*a(n) = k!*m = b(n), where k! is the largest factorial that divides b(n).
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0 0 1! * 1 1! * 1
1 1 2! * 1 2! * 1
2 2.1 3! * 2 3! * 2
3 3.1 4! * 1 4! * 1
4 4.2.1 5! * 6 6! * 1
5 5.2.1 6! * 2 6! * 2
6 6.3.1.1 7! * 12 7! * 12
7 7.3.1.1 8! * 3 8! * 3
8 8.4.2.1 9! * 10 10! * 1
9 9.4.2.1 10! * 2 10! * 2
10 10.5.2.1.1 11! * 12 12! * 1
11 11.5.2.1.1 12! * 2 12! * 2
12 12.6.3.2.1.1 13! * 420 15! * 2
13 13.6.3.2.1.1 14! * 60 15! * 4
14 14.7.3.2.1.1 15! * 24 15! * 24
15 15.7.3.2.1.1 16! * 3 16! * 3
16 16.8.4.2.1.1.1 17! * 90 18! * 5
...
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MATHEMATICA
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Table[j = 1; ( Times @@ Reap[While[Sow[#^Floor[n/(# - 1)]] &[Prime[j]] > 1, j++]][[-1, 1]] )/Factorial[n + 1], {n, 0, 60}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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