%I A005361 M0063
%S A005361 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,
%T A005361 3,2,1,1,1,5,1,1,1,4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,
%U A005361 1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,1,6,1,1,2,2,1,1
%N A005361 Product of exponents of prime factorization of n.
%C A005361 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) 
               since 24=2^3*3 and 375 =3*5^3 both have prime signature (3,1).
%C A005361 There was an old comment here that said "a(n) is the number of nilpotents 
               elements in the ring Z/nZ", but this is false - see A003557.
%C A005361 a(n) is the number of square-full divisors of n. a(n) is also the number 
               of divisors d of n such that d and n have the same prime factors, 
               i.e., A007947(d)=A007947(n). [From Laszlo Toth (ltoth(AT)ttk.pte.hu), 
               May 22 2009]
%D A005361 J. Knopfmacher, A prime-divisor function, Proc. Amer. Math. Soc., 40 
               (1973), 373-377. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), May 22 
               2009]
%D A005361 Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681.
%D A005361 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005361 D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full 
               divisors of an integer, Proc. Amer. Math. Soc., 34 (1972), 79-80. 
               [From Laszlo Toth (ltoth(AT)ttk.pte.hu), May 22 2009]
%H A005361 Daniel Forgues, <a href="b005361.txt">Table of n, a(n) for n=1..100000</
               a>
%F A005361 n = Product (p_j^k_j) -> a(n) = Product (k_j). Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/
               zeta(6s).
%F A005361 Multiplicative with a(p^e) = e. - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%F A005361 a(n)=Sum(d dividing n, floor(rad(d)/rad(n)), where rad(n) is A007947 
               [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 06 2009]
%t A005361 Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 
               2 ], 1 ]
%o A005361 (PARI) for(n=1,100,print1(prod(i=1,omega(n), component(component(factor(n),
               2),i)),","))
%Y A005361 Cf. A000005, A052306. a(p^k)=A000027=n. a(A002110)=A000012=1.
%Y A005361 Sequence in context: A157754 A072411 A091050 this_sequence A008479 A107345 
               A000688
%Y A005361 Adjacent sequences: A005358 A005359 A005360 this_sequence A005362 A005363 
               A005364
%K A005361 nonn,easy,nice,mult,new
%O A005361 1,4
%A A005361 Jeffrey Shallit, Olivier Gerard (olivier.gerard(AT)gmail.com)