A238645 -id:A238645 - cp's OEIS frontend

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

Zhi-Wei Sun, Mar 02 2014

nonn

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.

			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).

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.

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

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}]

A238703 a(n) = |{0 < k < n: floor(k*n/3) is prime}|.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 3, 1, 3, 4, 0, 4, 2, 1, 3, 5, 0, 4, 4, 1, 4, 5, 0, 3, 4, 0, 3, 6, 0, 5, 4, 1, 6, 6, 0, 7, 4, 1, 5, 4, 0, 7, 6, 0, 8, 5, 0, 8, 7, 1, 6, 7, 0, 9, 9, 1, 9, 8, 0, 6, 7, 0, 7, 12, 0, 9, 7, 1, 11, 10, 0, 6, 8, 0, 7, 9, 0, 7, 12

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: If n > m > 0 with n not divisible by m, then floor(k*n/m) is prime for some 0 < k < n.

			a(4) = 1 since floor(2*4/3) = 2 is prime.
If p is a prime, then a(3*p) = 1 since floor(k*3p/3) = k*p is prime only for k = 1. If m > 1 is composite, then a(3*m) = 0 since floor(k*3m/3) = k*m is composite for all k > 0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A237578, A237615, A238645, A238701.

p[n_,k_]:=PrimeQ[Floor[k*n/3]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

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

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