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

Author

M. F. Hasler, Nov 17 2014

Keywords

• nonn
• more

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.

Crossrefs

Cf. A000225, A005117, A136556.

Programs

Mathematica

Select[Range[0,57],SquareFreeQ[Binomial[2^#-1,#]]&] (* Ivan N. Ianakiev, Nov 18 2014 *)

PARI

is(n)=issquarefree(binomial(2^n-1,n))

PARI

is(n)={t=1; n&&n=vecsort(concat(vector(n,k,factor(if(kx)(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.