A057062 -id:A057062 - 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

A057045 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; the n-th Lucas number is in antidiagonal a(n).

Original entry on oeis.org

2, 1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 41, 52, 66, 85, 107, 137, 174, 221, 281, 358, 455, 579, 737, 937, 1192, 1516, 1929, 2454, 3121, 3970, 5050, 6424, 8171, 10394, 13221, 16818, 21393, 27212

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A057062, A057048, A022846, A057057, A057054.

from gmpy2 import isqrt_rem, lucas
def A057045(n):
    i, j = isqrt_rem(2*lucas(n-1))
    return int(i + int(4*(j-i) >= 1)) # Chai Wah Wu, Aug 16 2016

Round(sqrt(2*A000032(n-1))). - Vladeta Jovovic, Jun 14 2003

A057054 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; n^3 is in antidiagonal a(n).

Original entry on oeis.org

1, 4, 7, 11, 16, 21, 26, 32, 38, 45, 52, 59, 66, 74, 82, 91, 99, 108, 117, 126, 136, 146, 156, 166, 177, 187, 198, 210, 221, 232, 244, 256, 268, 280, 293, 305, 318, 331, 344, 358, 371, 385, 399, 413, 427, 441, 456, 470, 485, 500

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

Cf. A057045, A057062, A057048, A022846, A057057.

Table[Round[n Sqrt[2n]],{n,50}] (* Harvey P. Dale, Feb 10 2020 *)

Round(n*(sqrt(2*n))). - Vladeta Jovovic, Jun 14 2003

A057057 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; C(n,3) is in antidiagonal a(n).

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 13, 15, 18, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 55, 60, 64, 68, 72, 76, 81, 85, 90, 95, 100, 104, 109, 114, 119, 125, 130, 135, 141, 146, 152, 157, 163, 168, 174, 180, 186, 192, 198, 204, 210, 216, 223, 229

Offset: 3

Clark Kimberling, Jul 30 2000

nonn

Cf. A057045, A057062, A057048, A022846, A057054.

Table[Round[Sqrt[2*Binomial[n,3]]],{n,3,60}] (* Harvey P. Dale, Nov 16 2021 *)

Round(sqrt(2*binomial(n, 3))). - Vladeta Jovovic, Jun 14 2003

A123387 Number of triangular numbers <= n-th prime.

Original entry on oeis.org

2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27

Offset: 1

Giovanni Teofilatto, Nov 10 2006

Variant of A057062. - R. J. Mathar, Dec 13 2008

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000217, A002024, A003056.

f[n_] := Round[Sqrt[2n + 2]]; Table[f[Prime[n]], {n, 90}] (* Ray Chandler, Nov 13 2006 *)

from math import isqrt
from sympy import prime
def A123387(n): return isqrt(prime(n)+1<<3)+1>>1 # Chai Wah Wu, Oct 18 2022

a(n) = A003056(A000040(n)) + 1 = A002024(A000040(n) + 1).

a(n) ~ sqrt(2n log n). - Charles R Greathouse IV, Oct 18 2022

Extended by Ray Chandler, Nov 13 2006

cp's OEIS Frontend

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

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A057045 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; the n-th Lucas number is in antidiagonal a(n).

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A057054 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; n^3 is in antidiagonal a(n).

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A057057 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; C(n,3) is in antidiagonal a(n).

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A123387 Number of triangular numbers <= n-th prime.

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