A065382 -id:A065382 - 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

A111208 Number of primes <= n-th triangular number.

Original entry on oeis.org

0, 0, 2, 3, 4, 6, 8, 9, 11, 14, 16, 18, 21, 24, 27, 30, 32, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 74, 79, 84, 90, 94, 99, 102, 108, 114, 121, 126, 131, 137, 141, 149, 154, 160, 166, 174, 180, 188, 193, 200, 205, 216, 220, 226, 235, 242, 250, 259, 267, 274, 281, 290

Offset: 0

Giovanni Teofilatto, Oct 25 2005

nonn

Only because of the case n = 2 is it necessary to say "<=", otherwise "<" would suffice. Except for the first two terms, there are no consecutive identical terms for n < 10000. A065382 gives differences between consecutive terms of this sequence. - Alonso del Arte, Oct 31 2005

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000720, A000217.

Cf. A038107, A014085, A194189.

a111208 n = length $ takeWhile (<= a000217 n) a000040_list
-- Reinhard Zumkeller, Nov 01 2011

Table[PrimePi[n*(n + 1)/2], {n, 0, 60}] (* Ray Chandler, Oct 31 2005 *)

{ allocatemem(932245000); default(primelimit, 4294965247); write("b111208.txt", 0, " ", 0); for (n = 1, 10000, t=n*(n + 1)/2; a=primepi(t); write("b111208.txt", n, " ", a); ) } \\ Harry J. Smith, Mar 10 2009

[prime_pi(binomial(n,2)) for n in range(1, 63)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000217(n)).

Extended by Ray Chandler and Alonso del Arte, Oct 31 2005

A065383 a(n) = smallest prime >= n*(n + 1)/2.

Original entry on oeis.org

2, 2, 3, 7, 11, 17, 23, 29, 37, 47, 59, 67, 79, 97, 107, 127, 137, 157, 173, 191, 211, 233, 257, 277, 307, 331, 353, 379, 409, 439, 467, 499, 541, 563, 599, 631, 673, 709, 743, 787, 821, 863, 907, 947, 991, 1039, 1087, 1129, 1181, 1229, 1277, 1327, 1381, 1433

Offset: 0

Reinhard Zumkeller, Nov 05 2001

nonn

Besides 7, terms exclude the greater of the twin primes (A006512). - Bill McEachen, Dec 01 2022

Harry J. Smith, Table of n, a(n) for n = 0..1000

See A097050 for another version.
Cf. A065382, A007491, A064384, A000217.
Cf. A000217.

a065383 n = head $ dropWhile (< a000217 n) a000040_list
-- Reinhard Zumkeller, Aug 03 2012

PrimeNext[n_]:=Module[{k=n},While[ !PrimeQ[k],k++ ];k];f[n_]:=n*(n+1)/2;lst={};Do[AppendTo[lst,PrimeNext[f[n]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
NextPrime/@(Accumulate[Range[0,60]]-1) (* Harvey P. Dale, Jul 31 2012 *)

{ for (n=0, 1000, write("b065383.txt", n, " ", nextprime(n*(n + 1)/2)) ) } \\ Harry J. Smith, Oct 17 2009

Edited by N. J. A. Sloane, Nov 21 2008

A090970 Number of primes strictly between T(n) and T(n+1), where T(n) = n-th triangular number.

Original entry on oeis.org

1, 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: 1

Amarnath Murthy, Jan 03 2004

nonn

			a(8)=3 because between T(8)=36 and T(9)=45 we have the prime numbers 37,41 and 43.

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

Essentially the same as A065382 and A066888.

a:= proc(n) local ct,j: ct:=0: for j from n*(n+1)/2+1 to (n+1)*(n+2)/2-1 do if isprime(j)=true then ct:=ct+1 else ct:=ct fi: od: end: seq(a(n),n=1..103); # Emeric Deutsch, Feb 23 2005

With[{trs=Partition[Accumulate[Range[100]],2,1]},Join[{1},Rest[ PrimePi[ #[[2]]]- PrimePi[#[[1]]]&/@trs]]] (* Harvey P. Dale, Aug 25 2015 *)

from sympy import primerange
def A090970(n): return sum(1 for p in primerange((n*(n+1)>>1)+1,(n+2)*(n+1)>>1)) # Chai Wah Wu, May 22 2025

More terms from Emeric Deutsch, Feb 23 2005

A065384 Largest prime <= n * (n + 1) / 2.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 43, 53, 61, 73, 89, 103, 113, 131, 151, 167, 181, 199, 229, 251, 271, 293, 317, 349, 373, 401, 433, 463, 491, 523, 557, 593, 619, 661, 701, 739, 773, 811, 859, 887, 941, 983, 1033, 1069, 1123, 1171, 1223, 1259, 1321, 1373, 1429, 1483

Offset: 2

Reinhard Zumkeller, Nov 05 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A065382, A053001, A064383, A000217.

PrimePrev[n_]:=Module[{k=n},While[ !PrimeQ[k],k-- ];k];f[n_]:=n*(n+1)/2;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)

{ for (n=2, 1000, write("b065384.txt", n, " ", precprime(n*(n + 1)/2)) ) } [Harry J. Smith, Oct 17 2009]

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A111208 Number of primes <= n-th triangular number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Sage

Formula

Extensions

A065383 a(n) = smallest prime >= n*(n + 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A090970 Number of primes strictly between T(n) and T(n+1), where T(n) = n-th triangular number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A065384 Largest prime <= n * (n + 1) / 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula