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%I #6 Aug 16 2019 00:11:12
%S 1,3,7,8,14,24,37,65,97,105,145,163,253,686,1061,1871,2025,15255,
%T 28092,36183,56485,81294,81993,173338,328432,557890
%N Indices k of highly composite numbers with records of low values of the ratio between consecutive terms, A002182(k+1)/A002182(k).
%C Ramanujan proved that the asymptotic limit of the ratio between consecutive highly composite numbers is 1. Therefore this sequence is infinite.
%C The first 26 terms were calculated from Achim Flammenkamp's list of the first 779674 highly composite numbers.
%C 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, ...
%C 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, ...
%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly Composite Numbers</a>.
%H Srinivasa Ramanujan, <a href="https://doi.org/10.1112/plms/s2_14.1.347">Highly composite numbers</a>, Proceedings of the London Mathematical Society, Series 2, Vol. 14, No. 1 (1915), pp. 347-409, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram15.html">alternative link</a>.
%e The first 3 terms of the sequence are 1, 3, 7. A002182(1+1)/A002182(1) = 2/1 = 2, A002182(3+1)/A002182(3) = 6/4 = 3/2, A002182(7+1)/A002182(7) = 48/36 = 4/3, ... and 2 > 3/2 > 4/3 > ...
%t 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
%Y Cf. A002182.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Aug 15 2019
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