A268045 Least number k > 1 such that C(n+k,n) is squarefree.

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

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