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A005361 Product of exponents of prime factorization of n.
(Formerly M0063)
48
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, 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, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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).

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.

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

Number of u such that u^n|n|u. Row lengths in triangle of A284318. - Juri-Stepan Gerasimov, Apr 05 2017

REFERENCES

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

LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)

Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.

P. Erdős, T. Motzkin, Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect solution!)

J. Knopfmacher, A prime-divisor function, Proc. Amer. Math. Soc., 40 (1973), 373-377.

H. N. Shapiro, Problem 5735, Amer. Math. Monthly, 97 (1990), 937.

D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc., 34 (1972), 79-80.

Index entries for sequences computed from exponents in factorization of n

FORMULA

n = Product (p_j^k_j) -> a(n) = Product (k_j). Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).

Multiplicative with a(p^e) = e. - David W. Wilson, Aug 01 2001

a(n) = Sum_{d dividing n} floor(rad(d)/rad(n)) where rad(n) is A007947. - Enrique Pérez Herrero, Nov 06 2009

For n > 1: a(n) = Product_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011

a(n) = tau(n/rad(n)), where tau is A000005 and rad is A007947. - Anthony Browne, May 11 2016

a(n) = Sum_{k=1..n}(floor(cos^2(Pi*k^n/n))*floor(cos^2(Pi*n/k))). - Anthony Browne, May 11 2016

From Antti Karttunen, Mar 06 2017: (Start)

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

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

(End)

MAPLE

A005361 := proc(n)

    local a ;

    if n = 1 then

        1 ;

    else

        a := 1 ;

        for p in ifactors(n)[2] do

            a := a*op(2, p) ;

        end do:

    end if;

end proc:

seq(A005361(n), n=1..30) ; # R. J. Mathar, Nov 20 2012

MATHEMATICA

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

Array[Times@@Transpose[FactorInteger[#]][[2]]&, 80] (* Harvey P. Dale, Aug 15 2012 *)

PROG

(PARI) for(n=1, 100, print1(prod(i=1, omega(n), component(component(factor(n), 2), i)), ", "))

(PARI) a(n)=factorback(factor(n)[, 2]) \\ Charles R Greathouse IV, Nov 07 2014

(Haskell)

a005361 = product . a124010_row -- Reinhard Zumkeller, Jan 09 2012

(Scheme) (define (A005361 n) (if (= 1 n) 1 (* (A067029 n) (A005361 (A028234 n))))) ;; Antti Karttunen, Mar 06 2017

CROSSREFS

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

Sequence in context: A290107 A212180 A091050 * A008479 A227350 A107345

Adjacent sequences:  A005358 A005359 A005360 * A005362 A005363 A005364

KEYWORD

nonn,easy,nice,mult,changed

AUTHOR

Jeffrey Shallit, Olivier Gérard

STATUS

approved

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Last modified September 20 16:05 EDT 2017. Contains 292276 sequences.