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A238744
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Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.
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4
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1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2
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OFFSET
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2,6
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COMMENTS
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If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.
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LINKS
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FORMULA
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EXAMPLE
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24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
Rows begin:
1: () 16: (1,1,1,1) 31: (1)
2: (1) 17: (1) 32: (1,1,1,1,1)
3: (1) 18: (2,1) 33: (2)
4: (1,1) 19: (1) 34: (2)
5: (1) 20: (2,1) 35: (2)
6: (2) 21: (2) 36: (2,2)
7: (1) 22: (2) 37: (1)
8: (1,1,1) 23: (1) 38: (2)
9: (1,1) 24: (2,1,1) 39: (2)
10: (2) 25: (1,1) 40: (2,1,1)
11: (1) 26: (2) 41: (1)
12: (2,1) 27: (1,1,1) 42: (3)
13: (1) 28: (2,1) 43: (1)
14: (2) 29: (1) 44: (2,1)
15: (2) 30: (3) 45: (2,1)
(End)
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MATHEMATICA
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Table[Length/@Table[Select[Last/@FactorInteger[n], #>=k&], {k, Max@@Last/@FactorInteger[n]}], {n, 2, 100}] (* Gus Wiseman, Mar 31 2022 *)
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CROSSREFS
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These partitions are ranked by A238745.
A008480 gives number of permutations of prime indices, conjugate A321648.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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