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 A309745 Indices k of highly composite numbers with records of low values of the ratio between consecutive terms, A002182(k+1)/A002182(k). 0
 1, 3, 7, 8, 14, 24, 37, 65, 97, 105, 145, 163, 253, 686, 1061, 1871, 2025, 15255, 28092, 36183, 56485, 81294, 81993, 173338, 328432, 557890 (list; graph; refs; listen; history; text; internal format)
 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 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 > ... 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 Cf. A002182. Sequence in context: A127441 A067064 A093722 * A002381 A131559 A051211 Adjacent sequences:  A309742 A309743 A309744 * A309746 A309747 A309748 KEYWORD nonn,more AUTHOR Amiram Eldar, Aug 15 2019 STATUS approved

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Last modified December 5 17:42 EST 2019. Contains 329768 sequences. (Running on oeis4.)