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A268045 Least number k > 1 such that C(n+k,n) is squarefree. 1
2, 2, 2, 2, 2, 2, 4, 4, 3, 2, 2, 2, 2, 2, 9, 8, 3, 6, 2, 2, 2, 2, 36, 36, 20, 18, 36, 2, 2, 2, 16, 16, 2, 2, 3, 12, 2, 2, 4, 4, 2, 2, 2, 4, 3, 2, 16, 896, 175, 10, 2, 2, 9, 9, 4, 4, 2, 2, 2, 2, 2, 256, 417, 32, 2, 2, 2, 2, 2, 2, 4, 36, 2, 5, 5, 2, 2, 2, 81, 136, 135, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

By theorem 6 of the Granville-Ramaré link, a(n) exists for all n. - Robert Israel, Mar 01 2016

LINKS

Robert Israel, Table of n, a(n) for n = 0..477

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

MAPLE

F:= proc(n)

    local F, k;

    if numtheory:-issqrfree((n+2)*(n+1)/2) then return 2 fi;

    F:= ifactor((n+2)*(n+1)/2);

    for k from 3 do

      F:= F * ifactor(n+k)/ifactor(k);

      if not hastype(F, `^`) then return k fi

    od:

end proc:

map(F, [$0..100]); # Robert Israel, Jan 26 2016

MATHEMATICA

Table[k = 2; While[! SquareFreeQ@ Binomial[n + k, n], k++]; k, {n, 0, 81}] (* Michael De Vlieger, Jan 27 2016 *)

PROG

(PARI) findk(n) = {my(k=2); while (! issquarefree(binomial(n+k, n)), k++); k; } \\ Michel Marcus, Jan 26 2016

(PARI) b(n, p)=my(s); while(n\=p, s+=n); s

ok(n, k)=forprime(p=2, sqrtint(n+k), if(b(n+k, p)-b(k, p)-b(n, p)>1, return(0))); 1

a(n)=my(k=1); while(!ok(n, k++), ); k \\ Charles R Greathouse IV, Feb 18 2016

(Python)

from __future__ import division

from collections import Counter

from sympy import factorint

def A268045(n):

    if n == 0:

        return 2

    flist, k = Counter(factorint((n+2)*(n+1)//2)), 2

    while max(flist.values()) >= 2:

        k += 1

        flist += Counter(factorint(n+k))

        flist -= Counter(factorint(k))

    return k # Chai Wah Wu, Feb 17 2016

CROSSREFS

Sequence in context: A168514 A326353 A060447 * A118177 A105069 A172008

Adjacent sequences:  A268042 A268043 A268044 * A268046 A268047 A268048

KEYWORD

nonn

AUTHOR

Gionata Neri, Jan 25 2016

STATUS

approved

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Last modified May 16 12:54 EDT 2021. Contains 343947 sequences. (Running on oeis4.)