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A006255
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Ron Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1*b_2*...*b_t is a perfect square.
(Formerly M4064)
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4
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1, 6, 8, 4, 10, 12, 14, 15, 9, 18, 22, 20, 26, 21, 24, 16, 34, 27, 38, 30, 28, 33, 46, 32, 25, 39, 35, 40, 58, 42, 62, 45, 44, 51, 48, 36, 74, 57, 52, 50, 82, 56, 86, 55, 60, 69, 94, 54, 49, 63, 68, 65, 106, 70, 66, 72, 76, 87, 118, 75, 122, 93, 77, 64, 78, 80, 134, 85, 92, 84
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Every nonprime appears exactly once in this sequence.
If n is a square we can take t=1 and a(n) = n. If n is a prime > 3, then a(n) = 2n and t=3. If n is twice a prime, say p, then a(n) = 3p most of the time. The sequence b_1 < b_2 < ... < b_t will not contain either perfect squares or primes for they bring nothing to the solution. Also I know of no n such that t = 2. - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 30 2002
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REFERENCES
| R. L. Graham, Bijection between integers and composites, Problem 1242, Math. Mag., 60 (1980), 180.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 147.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| If n is a square we can take t=1 and a(n)=n.
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EXAMPLE
| a(2) = 6 because the best such sequence is 2,3,6. For n = 3 through 6 the {smallest m then smallest t then smallest product} solutions are 3,6,8; 4; 5,8,10; 6,8,12.
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CROSSREFS
| Having minimized m, next minimize t, then minimize product: A066400 and A066401 give values of t and square root of b_1*...*b_t.
Sequence in context: A200133 A021150 A065166 * A110760 A050710 A123092
Adjacent sequences: A006252 A006253 A006254 * A006256 A006257 A006258
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 30 2002
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