A109098
Numbers n such that n! is the product of exactly 5 smaller factorials (greater than 1).
Original entry on oeis.org
16, 48, 144, 192, 432, 576, 960, 1296, 1728, 2304, 2880, 5184, 5760, 6912, 8640, 11520, 17280, 20736, 25920, 27648, 34560, 40320, 51840, 57600, 69120, 82944
Offset: 1
144! = 2! * 2! * 3! * 3! * 143!, so 144 is in the sequence.
A109100
Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).
Original entry on oeis.org
10, 16, 24, 48, 96, 144, 192, 288, 384, 720, 768, 864, 1440, 1536, 1728, 3072, 4320, 5184, 6144, 8640, 10368, 12288, 24576, 25920, 31104, 40320, 49152, 51840, 62208, 80640, 86400, 98304
Offset: 1
10! = 3! * 5! * 7! = 6! * 7!, so 10 is in the sequence.
Cf.
A034878,
A001013,
A003135,
A058295,
A075082,
A109095,
A109096,
A109097,
A109098,
A109100,
A109101,
A109102,
A109103.
A109101
Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).
Original entry on oeis.org
576, 1152, 2304, 2880, 3456, 4608, 5760, 6912, 9216, 11520, 18432, 20736, 23040, 36864, 41472, 46080, 73728, 92160
Offset: 1
Cf.
A034878,
A001013,
A003135,
A058295,
A075082,
A109095,
A109096,
A109097,
A109098,
A109099,
A109100,
A109102,
A109103.
A109102
Numbers n such that n! is the product of exactly 4 smaller factorials (greater than 1).
Original entry on oeis.org
13824, 17280, 27648, 34560, 55296, 82944
Offset: 1
Cf.
A034878,
A001013,
A003135,
A058295,
A075082,
A109095,
A109096,
A109097,
A109098,
A109099,
A109100,
A109101,
A109103.
A109099
Numbers n such that n! can be expressed as the product of smaller factorials > 2.
Original entry on oeis.org
6, 10, 24, 36, 120, 144, 216, 576, 720, 864, 1296, 2880, 3456, 4320, 5040, 5184, 7776, 13824, 14400, 17280, 20736, 25920, 30240, 31104, 40320, 46656, 69120, 82944, 86400
Offset: 1
86400! = 5! * 6! * 86399!, so 86400 is in the sequence.
A109103
Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).
Original entry on oeis.org
4, 9, 288, 34560
Offset: 2
34560! = 2! * 3! * 4! * 5! * 34559!, using five different factorials, so a(5)=34560.
Cf.
A034878,
A001013,
A003135,
A058295,
A075082,
A109095,
A109096,
A109097,
A109098,
A109099,
A109100,
A109101,
A109102.
A336497
Numbers that cannot be written as a product of superfactorials A000178.
Original entry on oeis.org
3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76
Offset: 1
The sequence of terms together with their prime indices begins:
3: {2} 22: {1,5} 39: {2,6}
5: {3} 23: {9} 40: {1,1,1,3}
6: {1,2} 25: {3,3} 41: {13}
7: {4} 26: {1,6} 42: {1,2,4}
9: {2,2} 27: {2,2,2} 43: {14}
10: {1,3} 28: {1,1,4} 44: {1,1,5}
11: {5} 29: {10} 45: {2,2,3}
13: {6} 30: {1,2,3} 46: {1,9}
14: {1,4} 31: {11} 47: {15}
15: {2,3} 33: {2,5} 49: {4,4}
17: {7} 34: {1,7} 50: {1,3,3}
18: {1,2,2} 35: {3,4} 51: {2,7}
19: {8} 36: {1,1,2,2} 52: {1,1,6}
20: {1,1,3} 37: {12} 53: {16}
21: {2,4} 38: {1,8} 54: {1,2,2,2}
A006939 lists superprimorials or Chernoff numbers.
A303279 counts prime factors (with multiplicity) of superprimorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf.
A000142,
A000720,
A007489,
A011371,
A022559,
A022915,
A027423,
A034878,
A034876,
A076954,
A115627,
A294068.
-
supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[100],facsusing[Rest[Array[supfac,30]],#]=={}&]
A109104
Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.
Original entry on oeis.org
4, 6, 8, 9, 10, 12, 24, 32, 48, 72, 128, 192, 240, 384, 432, 480, 720, 864, 1152, 1440, 2592, 2880, 5040, 6144, 6912, 8192, 10080, 11520, 15360, 15552, 23040, 25920, 27648, 51840, 62208, 69120, 73728, 86400
Offset: 1
10! = 3! * 5! * 7!, so 10 is in the sequence.
Cf.
A034878,
A001013,
A003135,
A058295,
A075082,
A109095,
A109096,
A109097,
A109098,
A109099,
A109100,
A109101,
A109102,
A109103.
A363492
Numbers k such that the partition number p(k) = A000041(k) can be written as a product of smaller partition numbers.
Original entry on oeis.org
0, 1, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 39
Offset: 1
0 and 1 are terms, because p(0) = p(1) = 1 is the empty product.
7 is a term, because p(7) = 15 = 3*5 = p(3)*p(4).
39 is a term, because p(39) = 31185 = 3^4*385 = p(3)^4*p(18).
33 is not a term, even though all prime factors of p(33) = 3^2 * 7^2 * 23 appear in smaller partition numbers. (In particular, 33 is a term of A194345.) This is because the only smaller partition number that is divisible by 23 is p(32) = 3 * 11^2 * 23, but p(33) is not divisible by 11.
Except for a(1) = 0, subsequence of
A194345.
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