A181806
Positive integers with more highly composite divisors (A002182) than any smaller positive integer.
Original entry on oeis.org
1, 2, 4, 12, 24, 48, 120, 240, 360, 720, 5040, 10080, 15120, 30240, 60480, 151200, 166320, 332640, 665280, 1663200, 1995840, 3326400, 8648640, 17297280, 21621600, 43243200, 86486400, 129729600, 259459200, 735134400
Offset: 1
12 has five divisors (namely, 1, 2, 4, 6 and 12) that are members of A002182. No positive integer smaller than 12 has more than three members of A002182 among its divisors; hence, 12 is a member of the sequence.
A181807(n) = number of highly composite divisors of a(n) (i.e.,
A181801(a(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.
Original entry on oeis.org
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
Offset: 1
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.
- 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.
A212165 also includes all terms in
A006939,
A066120,
A087980,
A130091,
A138534,
A141586,
A166475,
A181555,
A181813-
A181814,
A181818,
A181823-
A181825,
A182763.
-
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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