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A003557 n divided by largest squarefree divisor of n. 48
1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n) is the size of the Frattini subgroup of the cyclic group C_n - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001. Also of the Frattini subgroup of the dihedral group with 2*n elements. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002

Number of solutions to x^m==0 (mod n) provided that n < 2^(m+1), i.e. the sequence of sequences A000188, A000189, A000190, etc. converges to this sequence. - Henry Bottomley, Sep 18 2001

a(n) is the number of nilpotent elements in the ring Z/nZ. - Laszlo Toth, May 22 2009

The sequence of partial products of a(n) is A085056(n). - Peter Luschny, Jun 29 2009

The first occurrence of n in this sequence is at A064659(n). - Franklin T. Adams-Watters, Jul 25 2014

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

H. Bottomley, Some Smarandache-type multiplicative sequences

S. R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006.

FORMULA

Multiplicative with a(p^e) = p^(e-1). - Vladeta Jovovic, Jul 23 2001

a(n) = n/rad(n) = n/A007947(n) = sqrt(J_2(n)/J_2(rad(n))), where J_2(n) is A007434. - Enrique Pérez Herrero, Aug 31 2010

a(n)=(J_k(n)/J_k(rad(n)))^(1/k), where J_k is the k-th Jordan Totient Function: (J_2 is A007434 and J_3 A059376). - Enrique Pérez Herrero, Sep 03 2010

Dirichlet convolution of A000027 and A097945. - R. J. Mathar, Dec 20 2011

a(n) = A000010(n)/|A023900(n)|. - Eric Desbiaux, Nov 15 2013

a(n) = product(A027748(n,k)^(A124010(n,k)-1): k = 1..A001221(n)). - Reinhard Zumkeller, Dec 20 2013

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

a(n) = e^[Sum_{k=2..n} (floor(n/k)-floor((n-1)/k))*(1-A010051(k))*Mangoldt(k)] where Mangoldt is the Mangoldt function. - Anthony Browne, Jun 16 2016

MAPLE

A003557 := n -> n/ilcm(op(numtheory[factorset](n))):

seq(A003557(n), n=1..98); # Peter Luschny, Mar 23 2011

MATHEMATICA

Prepend[ Array[ #/Times@@(First[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 1 ] (* Olivier Gérard, Apr 10 1997 *)

PROG

(Sage) def A003557(n) : return n*mul(1/p for p in prime_divisors(n))

[A003557(n) for n in (1..98)] # Peter Luschny, Jun 10 2012

(Haskell)

a003557 n = product $ zipWith (^)

                      (a027748_row n) (map (subtract 1) $ a124010_row n)

-- Reinhard Zumkeller, Dec 20 2013

(PARI) a(n)=n/factorback(factor(n)[, 1]) \\ Charles R Greathouse IV, Nov 17 2014

(Python)

from sympy.ntheory.factor_ import core

from sympy import divisors

def a(n): return n/max(list(filter(lambda i: core(i) == i, divisors(n))))

print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, Apr 16 2017

CROSSREFS

Cf. A007947, A062378, A062379, A064549.

Sequence in context: A104445 A000189 A000190 * A073752 A128708 A087653

Adjacent sequences:  A003554 A003555 A003556 * A003558 A003559 A003560

KEYWORD

nonn,easy,mult,changed

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.