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A212184
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Row n of table gives exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.
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1
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0, 0, 2, 0, 2, 3, 2, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 3, 2, 4, 4, 5, 2, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6, 3, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6
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OFFSET
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1,3
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COMMENTS
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Row n of table represents second signature of A002182(n) (cf. A212172). The use of 0 in the table to represent squarefree highly composite numbers accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.
No row is repeated an infinite number of times in the table. The contrary to this would imply that at least one integer appeared in A212183 an infinite number of times - something that Ramanujan proved to be false (cf. Ramanujan link). It would be interesting to know if there is an upper bound on the number of times a row can appear.
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REFERENCES
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S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
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LINKS
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FORMULA
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EXAMPLE
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First rows read: 0; 0; 2; 0; 2; 3; 2,2; 4; 2; 3; 2,2; 4;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Only exponents that are 2 or greater appear in a number's second signature; therefore, 12's second signature is {2}. Since 12 = A002182(5), row 5 represents the second signature {2}.
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CROSSREFS
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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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