

A000705


nth superior highly composite number A002201(n) is product of first n terms of this sequence.
(Formerly M0423 N0162)


5



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;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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.


REFERENCES

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. 1157, Ed. G. H. Hardy et al., AMS Chelsea 2000.
S. Ramanujan, Ramanujan's Papers, pp. 1479, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

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), 534543.
S. Ramanujan, First 50 primes whose products form successive superior highly composite numbers
Eric Weisstein's World of Mathematics, Superior Highly Composite Number


MATHEMATICA

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 *)


CROSSREFS

Cf. A004490, A073751.
Sequence in context: A086418 A100761 A027748 * A073751 A108501 A166226
Adjacent sequences: A000702 A000703 A000704 * A000706 A000707 A000708


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Edited by T. D. Noe, Nov 01 2002


STATUS

approved



