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A245569
Numbers n such that binomial(2^n-1,n) is squarefree.
1
0, 1, 2, 3, 4, 6, 11, 12, 21, 28, 29, 31, 51, 54, 55, 57
OFFSET
1,3
COMMENTS
Motivated by the previously existing entry for the subsequence A246699 of squarefree terms in this sequence.
Conjectured to be finite. Next term, if it exists, is larger than 500.
MATHEMATICA
Select[Range[0, 57], SquareFreeQ[Binomial[2^#-1, #]]&] (* Ivan N. Ianakiev, Nov 18 2014 *)
PROG
(PARI) is(n)=issquarefree(binomial(2^n-1, n))
(PARI) is(n)={t=1; n&&n=vecsort(concat(vector(n, k, factor(if(k<n, numerator, x->x)(t=(2^n-k)/k/denominator(t)), 0)~))); for(i=2, #n, if(n[1, i]==n[1, i-1], n[2, i]+=n[2, i-1]; n[2, i-1]=0)); !n||vecmax(n[2, ])<2} \\ Limiting "factor" to precomputed primes might yield a false positive, but no term > 57 is found so far.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
M. F. Hasler, Nov 17 2014
STATUS
approved