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%I M0423 N0162 #30 Dec 22 2021 00:10:45
%S 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,
%T 59,2,11,61,3,67,71,73,79,13,83,89,2,97,101,103,107,7,109,113,17,127,
%U 131,137,139,3,5,149,151,19,2,157,163,167,173,179,181,191,193,197,199
%N n-th superior highly composite number A002201(n) is product of first n terms of this sequence.
%C 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.
%D S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 115.
%D S. Ramanujan, Ramanujan's Papers, pp. 147-9, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000705/b000705.txt">Table of n, a(n) for n = 1..10000</a>
%H J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (#6, 2002), 534-543.
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page38.htm">First 50 primes whose products form successive superior highly composite numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperiorHighlyCompositeNumber.html">Superior Highly Composite Number</a>
%t 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 (* _T. D. Noe_, Nov 01 2002 *)
%Y Cf. A004490, A073751.
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_
%E Edited by _T. D. Noe_, Nov 01 2002
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