A238646 - OEIS

%I #14 Jul 10 2015 20:05:37

%S 0,0,0,1,2,2,2,2,3,1,2,1,3,1,3,1,4,2,3,2,5,4,5,1,3,3,4,2,5,3,4,5,8,3,

%T 5,1,5,5,7,3,5,2,6,3,6,6,9,4,8,7,7,6,7,4,6,7,8,5,6,4,7,8,9,6,6,6,9,5,

%U 7,4,8,6,10,6,5,8,11,7,10,6

%N Number of primes p < n such that the number of squarefree numbers among 1, ..., n-p is prime.

%C Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 10, 12, 14, 16, 24, 36.

%C This is analog of the conjecture in A237705 for squarefree numbers.

%C We have verified the conjecture for n up to 60000.

%H Zhi-Wei Sun, <a href="/A238646/b238646.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.

%e a(10) = 1 since 7 and 3 are both prime, and there are exactly 3 squarefree numbers among 1, ..., 10-7.

%e a(36) = 1 since 17 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., 36-17 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).

%t s[n_]:=Sum[If[SquareFreeQ[k],1,0],{k,1,n}]

%t a[n_]:=Sum[If[PrimeQ[s[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A005117, A013928, A237705, A237768, A237769, A238645.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Mar 02 2014