A238645 Number of odd primes p < 2*n such that the number of squarefree numbers among 1, ..., ((p-1)/2)*n is prime.

0, 1, 2, 1, 2, 2, 1, 2, 2, 5, 2, 3, 3, 1, 6, 5, 3, 3, 1, 4, 2, 4, 4, 3, 4, 2, 4, 3, 1, 4, 3, 3, 7, 5, 4, 5, 5, 4, 3, 2, 5, 2, 2, 4, 5, 4, 9, 7, 4, 3, 2, 4, 3, 4, 3, 2, 4, 6, 5, 6, 4, 4, 2, 2, 7, 5, 6, 6, 8, 3, 7, 3, 5, 6, 10, 6, 6, 6, 4, 5

Offset: 1

Author

Zhi-Wei Sun, Mar 02 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 4, 7, 14, 19, 29.

This is an analog of the conjecture in A237578 for squarefree numbers. We have verified it for n up to 20000.

See also A238646 for a similar conjecture involving squarefree numbers.

Examples

			a(4) = 1 since 3 is prime and there are exactly 3 squarefree numbers among 1, ..., (3-1)/2*4 (namely, 1, 2, 3).
a(14) = 1 since 5 and 17 are both prime, and there are exactly 17 squarefree numbers among 1, ..., (5-1)/2*14.
a(19) = 1 since 3 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., (3-1)/2*19 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).
a(29) = 1 since 41 and 353 are both prime, and there are exactly 353 squarefree numbers among 1, ..., (41-1)/2*29 = 580.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..900
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A005117, A013928, A237578, A237615, A238646.

Programs

Mathematica

s[n_]:=Sum[If[SquareFreeQ[k],1,0],{k,1,n}]
a[n_]:=Sum[If[PrimeQ[s[(Prime[k]-1)/2*n]],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]