OFFSET
1,2
COMMENTS
Ramanujan proved that the asymptotic limit of the ratio between consecutive highly composite numbers is 1. Therefore this sequence is infinite.
The first 26 terms were calculated from Achim Flammenkamp's list of the first 779674 highly composite numbers.
The corresponding highly composite numbers are A002182(a(n)) = 1, 4, 36, 48, 720, 25200, 665280, 698377680, 1606268664000, 8995104518400, 72779390658374400, ... and their corresponding consecutive terms are A002182(a(n)+1) = 2, 6, 48, 60, 840, 27720, 720720, 735134400, 1686582097200, 9316358251200, 74801040398884800, ...
The corresponding record ratios for the first 20 terms are of the form 1 + 1/m with m being an integer. The list of values of m is 1, 2, 3, 4, 6, 10, 12, 19, 20, 28, 36, 41, 176, 254, 345, 812, 9338, 10366, 21339, 44084, 89733/2, 497845/2, 435046, 800355, 30857708/23, 18882356170/7757, ...
LINKS
Achim Flammenkamp, Highly Composite Numbers.
Srinivasa Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Series 2, Vol. 14, No. 1 (1915), pp. 347-409, alternative link.
EXAMPLE
MATHEMATICA
s={}; hcn1 = 1; dm = 1; rm = 3; c=0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; hcn2 = n; c++; r = hcn2/hcn1; If[r < rm, rm = r; AppendTo[s, c]]; hcn1 = hcn2], {n, 2, 10^6}]; s
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Aug 15 2019
STATUS
approved