A005361 - OEIS

%I M0063 #141 Jan 28 2024 09:02:19

%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,

%T 1,4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,

%U 1,1,1,6,1,1,2,2,1,1

%N Product of exponents of prime factorization of n.

%C a(n) depends only on prime signature of n (cf. A025487, A052306). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

%C There was a comment here that said "a(n) is the number of nilpotents elements in the ring Z/nZ", but this is false, see A003557.

%C 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). - _Laszlo Toth_, May 22 2009

%C Number of divisors u of n such that u|(u^n/n). Row lengths in triangle of A284318. - _Juri-Stepan Gerasimov_, Apr 05 2017

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Daniel Forgues, <a href="/A005361/b005361.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from T. D. Noe)

%H Imanuel Chen and Michael Z. Spivey, <a href="http://soundideas.pugetsound.edu/summer_research/238">Integral Generalized Binomial Coefficients of Multiplicative Functions</a>, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.

%H P. Erdős and T. Motzkin, <a href="http://www.jstor.org/stable/2316593">Problem 5735</a>, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect solution!)

%H Vaclav Kotesovec, <a href="/A005361/a005361.jpg">Graph - the asymptotic ratio (100000 terms)</a>

%H J. Knopfmacher, <a href="http://dx.doi.org/10.1090/S0002-9939-1973-0327694-7">A prime-divisor function</a>, Proc. Amer. Math. Soc., 40 (1973), 373-377.

%H MathStackExchange, <a href="https://math.stackexchange.com/questions/4032015">Dirichlet series for zeta(s)*zeta(2*s)*zeta(3*s)*zeta(6*s)^(-1)</a>.

%H H. N. Shapiro, <a href="http://www.jstor.org/stable/2324350">Problem 5735</a>, Amer. Math. Monthly, 97 (1990), 937.

%H D. Suryanarayana and R. Sitaramachandra Rao, <a href="http://dx.doi.org/10.1090/S0002-9939-1972-0291104-8">The number of square-full divisors of an integer</a>, Proc. Amer. Math. Soc., Vol. 34, No. 1 (1972), pp. 79-80.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F n = Product (p_j^k_j) -> a(n) = Product (k_j).

%F Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).

%F Multiplicative with a(p^e) = e. - _David W. Wilson_, Aug 01 2001

%F a(n) = Sum_{d dividing n} floor(rad(d)/rad(n)) where rad(n) is A007947. - _Enrique Pérez Herrero_, Nov 06 2009

%F For n > 1: a(n) = Product_{k=1..A001221(n)} A124010(n,k). - _Reinhard Zumkeller_, Aug 27 2011

%F a(n) = tau(n/rad(n)), where tau is A000005 and rad is A007947. - _Anthony Browne_, May 11 2016

%F a(n) = Sum_{k=1..n}(floor(cos^2(Pi*k^n/n))*floor(cos^2(Pi*n/k))). - _Anthony Browne_, May 11 2016

%F From _Antti Karttunen_, Mar 06 2017: (Start)

%F For all n >= 1, a(prime^n) = n, a(A002110(n)) = a(A005117(n)) = 1. [From Crossrefs section.]

%F a(1) = 1; for n > 1, a(n) = A067029(n) * a(A028234(n)).

%F (End)

%F Let (b(n)) be multiplicative with b(p^e) = -1 + ( (floor((e-1)/3)+floor(e/3)) mod 4 ) for p prime and e > 0, then b(n) is the Dirichlet inverse of (a(n)). - _Werner Schulte_, Feb 23 2018

%F Sum_{i=1..k} a(i) ~ (zeta(2)*zeta(3)/zeta(6)) * k (Suryanarayana and Sitaramachandra Rao, 1972). - _Amiram Eldar_, Apr 13 2020

%F More precise asymptotics: Sum_{k=1..n} a(k) ~ 315*zeta(3)*n / (2*Pi^4) + zeta(1/2)*zeta(3/2)*sqrt(n) / zeta(3) + 6*zeta(1/3)*zeta(2/3)*n^(1/3) / Pi^2 [Knopfmacher, 1973]. - _Vaclav Kotesovec_, Jun 13 2020

%p A005361 := proc(n)

%p     local a, p ;

%p     a := 1 ;

%p     for p in ifactors(n)[2] do

%p        a := a*op(2, p) ;

%p     end do:

%p     a ;

%p end proc:

%p seq(A005361(n),n=1..30) ; # _R. J. Mathar_, Nov 20 2012

%p # second Maple program:

%p a:= n-> mul(i[2], i=ifactors(n)[2]):

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 18 2020

%t Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]

%t Array[Times@@Transpose[FactorInteger[#]][[2]]&,80] (* _Harvey P. Dale_, Aug 15 2012 *)

%o (PARI) for(n=1,100, f=factor(n); print1(prod(i=1,omega(f), f[i,2]),",")) \\ edited by _M. F. Hasler_, Feb 18 2020

%o (PARI) a(n)=factorback(factor(n)[,2]) \\ _Charles R Greathouse IV_, Nov 07 2014

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X + X^2)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020

%o (Haskell)

%o a005361 = product . a124010_row -- _Reinhard Zumkeller_, Jan 09 2012

%o (Scheme) (define (A005361 n) (if (= 1 n) 1 (* (A067029 n) (A005361 (A028234 n))))) ;; _Antti Karttunen_, Mar 06 2017

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def a(n): return prod(factorint(n).values())

%o print([a(n) for n in range(1, 91)]) # _Michael S. Branicky_, Jul 04 2022

%Y Cf. A000005, A002110, A005117 (indices of ones), A028234, A052306, A067029, A072411, A082695, A284318.

%Y Cf. A360969, A360970, A361132.

%Y Cf. A340065 (Dgf at s=2).

%K nonn,easy,nice,mult

%O 1,4

%A _Jeffrey Shallit_ and _Olivier Gérard_