A090970 -id:A090970 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 06 2003

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

			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)

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

Cf. A083382.
Essentially the same as A065382 and A090970.
Cf. A000217, A000040, A014085, A190661.

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 *)

{ 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; ) }

{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;) }

from sympy import primerange
def A066888(n): return sum(1 for p in primerange((n*(n+1)>>1)+1,((n+2)*(n+1)>>1)+1)) # Chai Wah Wu, May 22 2025

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

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

Offset corrected by Daniel Forgues, Sep 05 2012

A065382 Number of primes between n(n+1)/2 (exclusive) and (n+1)(n+2)/2 (inclusive).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 05 2001

nonn

Inspired by the weaker Legendre conjecture that there should be at least one prime between n^2 and (n+1)^2.

			a(10) = 2 because between 10*(10+1)/2=55 and (10+1)*(10+2)/2=66 there are 2 primes: 59, 61.

T. D. Noe, Table of n, a(n) for n = 1..10000

A000217, A014085, A065383, A065384.

Essentially the same as A066888 and A090970.

Table[ PrimePi[n(n + 1)/2] - PrimePi[n(n - 1)/2], {n, 2, 96}]

from sympy import primerange
def A065382(n): return sum(1 for p in primerange((n*(n+1)>>1)+1,((n+2)*(n+1)>>1)+1)) # Chai Wah Wu, May 22 2025

Definition improved by Robert G. Wilson v, Apr 22 2003

A282518 Number of n-element subsets of [n+1] having a prime element sum.

Original entry on oeis.org

0, 1, 2, 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

Offset: 0

Alois P. Heinz, Feb 17 2017

nonn

			a(1) = 1: {2}.
a(2) = 2: {1,2}, {2,3}.
a(3) = 1: {1,2,4}.
a(4) = 2: {1,2,3,5}, {1,3,4,5}.
a(5) = 2: {1,2,3,5,6}, {1,3,4,5,6}.
a(6) = 1: {1,2,3,4,6,7}.
a(7) = 2: {1,2,3,4,5,6,8}, {1,2,3,4,6,7,8}.
a(8) = 3: {1,2,3,4,5,6,7,9}, {1,2,3,5,6,7,8,9}, {1,3,4,5,6,7,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000720, A282516.

Similar but different: A065382, A066888, A090970.

a:= proc(n) option remember; (t-> add(`if`(isprime(
       t-i), 1, 0), i=1..n+1))((n+1)*(n+2)/2)
    end:
seq(a(n), n=0..100);

a(n) = A282516(n+1,n).

a(n) = pi((n+1)*(n+2)/2)-pi(n*(n+1)/2) for n >= 3, pi = A000720.

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

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A065382 Number of primes between n(n+1)/2 (exclusive) and (n+1)(n+2)/2 (inclusive).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Python

Extensions

A282518 Number of n-element subsets of [n+1] having a prime element sum.

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Author

Keywords

Examples

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Programs

Maple

Formula