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A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

%I A212169 #19 Jun 12 2022 22:51:54
%S A212169 1,2,4,12,24,36,48,120,240,360,720,1680,5040,10080,15120,20160,25200,
%T A212169 45360,50400,110880,221760,332640,554400,665280,2882880,8648640,
%U A212169 14414400,17297280,43243200,294053760
%N A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.
%C A212169 Sequence can be used to find the largest highly composite number in subsequences of A212165 (of which there are several in the database).
%C A212169 Ramanujan showed that, in the canonical prime factorization of a highly composite number with largest prime factor prime(n), the largest exponent cannot exceed 2*log_2(prime(n+1)). (See formula 54 on page 15 of the Ramanujan paper.) This limit is less than n for all n >= 9 (and prime(n) >= 23).
%C A212169 1. Direct calculation verifies this for 9 <= n <= 11.
%C A212169 2. Nagura proved that, for any integer m >= 25, there is always a prime between m and 1.2*m.  Let n = 11, at which point prime(11) = 31 and log_2(prime(n+1)) = log 37/log 2 = 5.209453.... Since log 1.2/log 2 is only 0.263034..., it follows that n must increase by at least 3k before 2*log_2(prime(n+1)) can increase by 2k, for all values of k. Therefore, 2*log_2(prime(n+1)) can never catch up to prime(n) for n > 11.
%C A212169 665280 = 2^6*3^3*5*7*11 is the largest highly composite number whose prime factorization contains an exponent that is strictly greater than the number of positive exponents in that factorization (including the implied 1's).
%D A212169 S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
%H A212169 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>
%H A212169 J. Nagura, <a href="http://projecteuclid.org/euclid.pja/1195570997">On the interval containing at least one prime number</a>, Proc. Japan Acad., 28 (1952), 177-181.
%H A212169 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page15.htm">Highly Composite Numbers (p. 15)</a> (pages 11-12 introduce some of the notation in formula 54)
%e A212169 A002182(62) = 294053760 = 2^7*3^3*5*7*11*13*17 has 7 positive exponents in its prime factorization, including 5 implied 1's. The maximal exponent in its prime factorization is also 7. Therefore, 294053760 is a term of this sequence.
%t A212169 okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; a = 0; t = {}; Do[b = DivisorSigma[0, n]; If[b > a, a = b; If[okQ[n], AppendTo[t, n]]], {n, 10^6}]; t (* _T. D. Noe_, May 24 2012 *)
%Y A212169 A212165 also includes all terms in A006939, A066120, A087980, A130091, A138534, A141586, A166475, A181555, A181813-A181814, A181818, A181823-A181825, A182763.
%Y A212169 Other finite subsequences of A002182 include A072938, A095921, A106037, A136253, A162935, A166981, A168263.
%K A212169 nonn,fini,full
%O A212169 1,2
%A A212169 _Matthew Vandermast_, May 22 2012