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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 29 12:15 EDT 2023. Contains 363042 sequences. (Running on oeis4.)