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A212184 Row n of table gives exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.

%I A212184 #22 Jan 20 2025 03:51:09
%S A212184 0,0,2,0,2,3,2,2,4,2,3,2,2,4,3,2,4,2,3,2,2,4,3,2,4,2,3,3,5,2,4,3,6,2,
%T A212184 4,2,2,3,2,4,4,5,2,2,4,2,3,3,5,2,4,3,6,2,4,2,2,5,3,4,4,5,2,2,6,3,4,2,
%U A212184 3,3,5,2,4,3,6,2,4,2,2,5,3,4,4,5,2,2,6
%N A212184 Row n of table gives exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.
%C A212184 Length of row n equals A212185(n) if A212185(n) is positive, or 1 if A212185(n) = 0.
%C A212184 Row n of table represents second signature of A002182(n) (cf. A212172). The use of 0 in the table to represent squarefree highly composite numbers accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.
%C A212184 No row is repeated an infinite number of times in the table.  The contrary to this would imply that at least one integer appeared in A212183 an infinite number of times - something that Ramanujan proved to be false (cf. Ramanujan link). It would be interesting to know if there is an upper bound on the number of times a row can appear.
%D A212184 Srinivasa 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 A212184 Amiram Eldar, <a href="/A212184/b212184.txt">Table of n, a(n) for n = 1..5594</a> (first 1000 rows)
%H A212184 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>.
%H A212184 Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page34.htm">Highly Composite Numbers</a> (p. 34).
%F A212184 Row n is identical to row A002182(n) of table A212172.
%e A212184 First rows read: 0; 0; 2; 0; 2; 3; 2,2; 4; 2; 3; 2,2; 4;...
%e A212184 12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Only exponents that are 2 or greater appear in a number's second signature; therefore, 12's second signature is {2}.  Since 12 = A002182(5), row 5 represents the second signature {2}.
%t A212184 With[{v = Import["https://oeis.org/A002182/b002182.txt", "Table"][[;; , 2]]}, exp[n_] := Select[FactorInteger[n][[;; , 2]], # > 1 &]; exp /@ v[[1 ;; 100]] /. {} -> {0} // Flatten] (* _Amiram Eldar_, Jan 20 2025 *)
%Y A212184 Cf. A002182, A212172, A212182, A212183, A212185.
%K A212184 nonn,tabf
%O A212184 1,3
%A A212184 _Matthew Vandermast_, Jul 01 2012