A245569 Numbers n such that binomial(2^n-1,n) is squarefree.

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.

Links

Table of n, a(n) for n=1..16.

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

Cf. A000225, A005117, A136556.

Sequence in context: A358788 A014859 A084166 * A018297 A050886 A341652

Adjacent sequences: A245566 A245567 A245568 * A245570 A245571 A245572

Keyword

nonn,more

Author

M. F. Hasler, Nov 17 2014

Status

approved