A000705 n-th superior highly composite number A002201(n) is product of first n terms of this sequence. (Formerly M0423 N0162)

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

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, Ramanujan's Papers, pp. 147-9, 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), 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

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: A027748 A361650 A328852 * A073751 A319431 A258581

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