A319506 Number of numbers of the form 2*p or 3*p between consecutive triangular numbers T(n - 1) < {2,3}*p <= T(n) with p prime.

0, 0, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 4, 5, 2, 4, 3, 5, 5, 2, 6, 3, 5, 5, 5, 5, 6, 4, 3, 7, 5, 6, 6, 5, 6, 7, 4, 5, 6, 6, 7, 6, 7, 9, 6, 6, 7, 8, 5, 6, 7, 9, 7, 7, 8, 7, 11, 7, 8, 8, 7, 6, 11, 5, 12, 7, 7, 7, 11, 11, 7, 12, 10, 9, 10, 7, 9, 9, 8, 10, 12, 10, 7, 10, 9, 12, 9, 11, 10, 13, 14, 10, 7

Offset: 1

Author

Ralf Steiner, Sep 21 2018

Keywords

• nonn

Comments

1) It is conjectured that for k >= 1 each left-sided half-open interval (T(2*k - 1), T(2*k + 1)] and (T(2*k), T(2*(k + 1))] contains at least one composite c_2 = 2*p_i and c_3 = 3*p_j each, p_i, p_j prime, i != j.

2) It is conjectured that for k >= 3 each left-sided half-open interval (T(k - 1), T(k)] contains at least one composite c_2 = 2*p_i or c_3 = 3*p_j, p_i, p_j prime, i != j.

3) It is conjectured that for k >= 2 each left-sided half-open interval (T(2*k - 1), T(2*k)] contains at least one composite c_3 = 3*p_j, p_j prime.

4) It is conjectured that for k >= 1 each left-sided half-open interval (T(2*k), T(2*k + 1)] contains at least one composite c_2 = 2*p_i, p_i prime.

Examples

			a(3) = 2 since (T(3 - 1),T(3)] = {4 = 2*2,5,6 = 2*3 = 3*2}, 2,3 prime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040 (primes), A000217 (triangular numbers).

Programs

Mathematica

Table[Count[
  Select[Range[(n - 1) n/2 + 1, n (n + 1)/2],
   PrimeQ[#/2] || PrimeQ[#/3] &], _Integer], {n, 1, 100}]
p23[{a_,b_}]:=Module[{r=Range[a+1,b]},Count[Union[Join[r/2,r/3]], ?PrimeQ]]; p23/@Partition[Accumulate[Range[0,100]],2,1] (* _Harvey P. Dale, May 02 2020 *)

PARI

isok1(n, k) = ((n%k) == 0) && isprime(n/k);
isok2(n) = isok1(n,2) || isok1(n,3);
t(n) = n*(n+1)/2;
a(n) = sum(i=t(n-1)+1, t(n), isok2(i)); \\ Michel Marcus, Oct 12 2018