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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

Cf. A004490, A073751.

Programs

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 *)

Extensions

Edited by T. D. Noe, Nov 01 2002