login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A066888 Number of primes p between triangular numbers T(n) < p <= T(n+1). 12
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 (list; graph; refs; listen; history; text; internal format)
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+2sqrt(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 *)

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 (zevi_35711(AT)yahoo.com), Jun 06 2003

Offset corrected by Daniel Forgues, Sep 05 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 25 04:48 EDT 2017. Contains 289779 sequences.