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A212169
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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.
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1
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1, 2, 4, 12, 24, 36, 48, 120, 240, 360, 720, 1680, 5040, 10080, 15120, 20160, 25200, 45360, 50400, 110880, 221760, 332640, 554400, 665280, 2882880, 8648640, 14414400, 17297280, 43243200, 294053760
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Sequence can be used to find the largest highly composite number in subsequences of A212165 (of which there are several in the database).
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).
1. Direct calculation verifies this for 9 <= n <= 11.
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.
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).
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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A212165 also includes all terms in A006939, A066120, A087980, A130091, A138534, A141586, A166475, A181555, A181813-A181814, A181818, A181823-A181825, A182763.
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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