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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 02 2014
STATUS
approved