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Search: id:A000705
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| A000705 |
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n-th superior highly composite number A002201(n) is product of first n terms of this sequence.
(Formerly M0423 N0162)
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+0
4
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| 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, 47, 5, 53, 59, 2, 11, 61, 3, 67, 71, 73, 79, 13, 83, 89, 2, 97, 101, 103, 107, 7, 109, 113, 17, 127, 131, 137, 139, 3, 5, 149, 151, 19, 2, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The Mathematica program uses the fact that the ratio of consecutive superior highly composite numbers is a prime, which was proved by Ramanujan. Ramanujan computed the first 50 terms of this sequence. Related sequences are A004490 and A073751, having to do with colossally abundant numbers.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 115.
S. Ramanujan, Collected Papers of Srinivasa Ramanujan, pp. 115-7, Ed. G. H. Hardy et al., AMS Chelsea 2000.
S. Ramanujan, Ramanujan's Papers, pp. 147-9, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
S. Ramanujan, First 50 primes whose products form successive superior highly composite numbers
Eric Weisstein's World of Mathematics, Superior Highly Composite Number
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MATHEMATICA
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pFactor[f_List] := Module[{p = f[[1]], k = f[[2]]}, N[Log[(k + 2)/(k + 1)]/Log[p]]]; maxN = 100; f = {{2, 1}, {3, 0}}; primes = 1; lst = {2}; x = Table[pFactor[f[[i]]], {i, primes + 1}]; For[n = 2, n <= maxN, n++, i = Position[x, Max[x]][[1, 1]]; AppendTo[lst, f[[i, 1]]]; f[[i, 2]]++; If[i > primes, primes++; AppendTo[f, {Prime[i + 1], 0}]; AppendTo[x, pFactor[f[[ -1]]]]]; x[[i]] = pFactor[f[[i]]]]; lst
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CROSSREFS
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Cf. A004490, A073751.
Sequence in context: A086418 A100761 A027748 this_sequence A073751 A108501 A166226
Adjacent sequences: A000702 A000703 A000704 this_sequence A000706 A000707 A000708
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Edited by T. D. Noe (noe(AT)sspectra.com), Nov 01 2002
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