A238646 Number of primes p < n such that the number of squarefree numbers among 1, ..., n-p is prime.

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

Offset

1,5

Comments

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

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

We have verified the conjecture for n up to 60000.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A283472 A225538 A212355 * A194330 A280667 A194286

Adjacent sequences: A238643 A238644 A238645 * A238647 A238648 A238649

Keyword

nonn

Author

Zhi-Wei Sun, Mar 02 2014

Status

approved