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A292441 Largest m such that m^2 divides A000984(n). 2
1, 1, 1, 2, 1, 6, 2, 2, 3, 2, 2, 2, 2, 10, 30, 12, 3, 6, 10, 10, 6, 2, 2, 60, 30, 42, 42, 28, 2, 4, 4, 4, 21, 14, 14, 6, 2, 2, 10, 140, 14, 126, 6, 60, 90, 12, 84, 84, 210, 30, 18, 12, 6, 36, 4, 4, 6, 4, 4, 12, 12, 132, 132, 440, 55, 330, 10, 10, 90, 30, 30, 180 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is the product of p^floor(m(n,p)/2) over primes p<n, where m(n,p) is the number of carries when adding n to itself in base p. - Robert Israel, Sep 17 2017

Granville and Ramaré show that A006530(a(n)) > sqrt(n/5) if n >= 2082.

In particular a(n) -> infinity as n -> infinity. - Robert Israel, Sep 18 2017

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].

Eric Weisstein's World of Mathematics, Erdős Squarefree Conjecture

Wikipedia, Kummer's theorem

FORMULA

a(n) > 1 for n > 4.

a(n) = A000188(A000984(n)). - Robert Israel, Sep 17 2017

EXAMPLE

binomial(10,5)/7           =  252/7   = 36 = a(5)^2.

binomial(12,6)/(3*7*11)    =  924/231 =  4 = a(6)^2.

binomial(14,7)/(2*3*11*13) = 3432/858 =  4 = a(7)^2.

MAPLE

A000188:= n -> mul(t[1]^floor(t[2]/2), t = ifactors(n)[2]):

seq(A000188(binomial(2*n, n)), n=0..100); # Robert Israel, Sep 17 2017

CROSSREFS

Cf. A000188, A000984, A006530, A292442.

Sequence in context: A157392 A134134 A222005 * A198870 A050457 A195441

Adjacent sequences:  A292438 A292439 A292440 * A292442 A292443 A292444

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 16 2017

STATUS

approved

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Last modified August 14 06:01 EDT 2018. Contains 313748 sequences. (Running on oeis4.)