A066888 Number of primes p between triangular numbers T(n) < p <= T(n+1).

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

Offset

0,2

Comments

It is conjectured that for n > 0, a(n) > 0. See also A190661. - John W. Nicholson, May 18 2011

If the above conjecture is true, then for any k > 1 there is a prime p > k such that p <= (n+1)(n+2)/2, where n = floor(sqrt(2k)+1/2). Ignoring the floor function we can obtain a looser (but nicer) lower bound of p <= 1 + k + 2*sqrt(2k). - Dmitry Kamenetsky, Nov 26 2016

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Formula

a(n) = pi(n*(n+1)/2) - pi(n*(n-1)/2).

a(n) equals the number of occurrences of n+1 in A057062. - Esko Ranta, Jul 29 2011

Example

Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of primes in n-th row.
Triangle begins
   1              (0 primes)
   2  3           (2 primes)
   4  5  6        (1 prime)
   7  8  9 10     (1 prime)
  11 12 13 14 15  (2 primes)

Mathematica

Table[PrimePi[(n^2 + n)/2] - PrimePi[(n^2 - n)/2], {n, 96}] (* Alonso del Arte, Sep 03 2011 *)
PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Partition[Accumulate[Range[0, 100]], 2, 1] (* Harvey P. Dale, Jun 04 2019 *)

Prog

(PARI) { tp(m)=local(r, t); r=1; for(n=1, m, t=0; for(k=r, n+r-1, if(isprime(k), t++)); print1(t", "); r=n+r; ) }
(PARI) {tpf(m)=local(r, t); r=1; for(n=1, m, t=0; for(k=r, n+r-1, if(isprime(k), t++); print1(k" ")); print1(" ("t" prime)"); print(); r=n+r; ) }

Crossrefs

Cf. A083382.

Essentially the same as A065382 and A090970.

Cf. A000217, A000040, A014085, A190661.

Sequence in context: A025848 A268197 A065382 * A029313 A144001 A124233

Adjacent sequences: A066885 A066886 A066887 * A066889 A066890 A066891

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2003

Extensions

More terms from Vladeta Jovovic and Jason Earls, Jun 06 2003

Offset corrected by Daniel Forgues, Sep 05 2012

Status

approved