A190661 -id:A190661 - cp's OEIS frontend

A190881 Exception list of where A190661(n) < A104272(n) for n > 0.

Original entry on oeis.org

43, 45, 68, 93, 145, 341, 655, 3177, 3383, 3424, 4696, 5109, 5116, 5183, 5201, 5225, 7195, 7574

Offset: 1

John W. Nicholson, May 23 2011

nonn

a(n) is the value of n of A190661 and A104272 at the exception.

Cf. A066888, A000217, A000040, A088634, A190661, A104272.

Corrected data for the changed offset and data of A190661 - John W. Nicholson, Nov 20 2013

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

Original entry on oeis.org

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

A191225 Number of Ramanujan primes R_k between triangular numbers T(n-1) < R_k <= T(n).

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 0, 2, 3, 2, 1, 2, 2, 2, 1, 4, 2, 2, 1, 0, 4, 3, 5, 1, 3, 2, 1, 5, 1, 2, 3, 4, 4, 4, 2, 2, 2, 4, 2, 3, 4, 3, 5, 4, 3, 2, 5, 4, 2, 5, 1, 6, 1, 5, 5, 7, 2, 2, 1, 10, 6, 6, 2, 2, 5, 0, 3, 7, 5, 4, 6, 7, 4

Offset: 1

John W. Nicholson, May 27 2011

nonn

The function eta(x), A191228, returns the greatest value of k of R_k <= x, and where R_k is the k-th Ramanujan prime (A104272).

			Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of Ramanujan primes in n-th row.
Triangle begins
1                 (0 Ramanujan primes, eta(1) = 0)
2  3              (1 Ramanujan primes, eta(3) - eta(1) = 1)
4  5  6           (0 Ramanujan primes, eta(6) - eta(3) = 0)
7  8  9  10       (0 Ramanujan primes, eta(10) - eta(6) = 0)
11 12 13 14 15    (1 Ramanujan primes, eta(15) - eta(10) = 1)
16 17 18 19 20 21 (1 Ramanujan primes, eta(21) - eta(15) = 1)

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A000217, A000040, A104272, A191228, A014085, A190661, A083382, A191226, A191227.

terms = 100; nn = terms^2; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s+1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
eta = Table[Boole[MemberQ[A104272, k]], {k, 1, nn}] // Accumulate;
T[n_] := n(n+1)/2;
a[1] = 0; a[n_] := eta[[T[n]]] - eta[[T[n-1]]];
Array[a, terms] (* Jean-François Alcover, Nov 07 2018, using T. D. Noe's code for A104272 *)

use ntheory ":all"; sub a191225 { my $n = shift; ramanujan_prime_count( (($n-1)*$n)/2+1, ($n*($n+1))/2 ); } say a191225($) for 1..10; # _Dana Jacobsen, Dec 30 2015

a(n) = eta(T(n))- eta(T(n-1)).

A088634 Index of the first occurrence of n in A066888.

Original entry on oeis.org

1, 3, 2, 9, 17, 28, 30, 36, 41, 54, 74, 51, 65, 92, 100, 112, 118, 132, 108, 154, 158, 161, 172, 175, 210, 197, 215, 255, 248, 239, 236, 316, 297, 291, 340, 321, 330, 345, 334, 400, 406, 402, 423, 394, 445, 452, 509, 493, 507, 481, 526, 546, 561, 584, 565, 598

Offset: 0

Amarnath Murthy, Oct 26 2003

For all 0 < n < 1000, a(n) < A104272(n). - John W. Nicholson, May 22 2011

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

Cf. A066888, A000217, A000040, A190661.

nn=100; k=0; t=Table[0,{nn}]; cnt=0; While[cntT. D. Noe, May 19 2011 *)

More terms from David Wasserman, Aug 16 2005

A191226 First occurrence of number n of Ramanujan primes in A191225.

Original entry on oeis.org

1, 2, 12, 22, 29, 36, 65, 69, 117, 118, 73, 100, 108, 154, 161, 200, 254, 172, 274, 239, 340, 321, 334, 330, 345, 471, 378, 481, 357, 526, 522, 515, 610, 635, 612, 655, 648, 792, 809, 731, 797, 594, 806, 851, 988, 886, 963, 933, 1005, 1111, 927, 1124, 970

Offset: 0

John W. Nicholson, May 28 2011

nonn

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A000217, A000040, A104272, A191228, A014085, A190661, A083382, A191225, A191227.

A191227 Last known occurrence of number n of Ramanujan primes in A191225.

Original entry on oeis.org

79, 194, 153, 284, 420, 333, 454, 592, 504, 412, 652, 512, 486, 617, 613, 660, 1130, 753, 1002, 849, 1060, 957, 1034, 1037, 1198, 961, 969, 1056, 1368, 1400, 1264, 1314, 1301, 1683, 1510, 1571, 1532, 1625, 1771, 1810, 1745, 1907, 1961, 1877, 1851, 2104, 2097

Offset: 0

John W. Nicholson, May 28 2011

nonn

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A000217, A000040, A104272, A191228, A014085, A190661, A083382, A191225, A191226.

A230147 Record values in A165959.

Original entry on oeis.org

2, 3, 5, 11, 15, 27, 33, 37, 65, 67, 75, 77, 95, 137, 147, 151, 153, 169, 191, 219, 247, 249, 251, 291, 297, 303, 307, 319, 415, 429, 441, 465, 495

Offset: 1

John W. Nicholson, Nov 20 2013

nonn

The index value A230146(n) is also the index of A104272(n).

Because of the bounds on a(n), A182873(n), A192820, and A190661, old conjectures about prime numbers may be proved.

Cf. A230146 = indices of record values of A165959.

mn=1; for(n=1, 10^8, if(A165959(n)>A165959(mn), {mn=n; print(mn, " ", A165959(mn))}))

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A190881 Exception list of where A190661(n) < A104272(n) for n > 0.

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

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A191225 Number of Ramanujan primes R_k between triangular numbers T(n-1) < R_k <= T(n).

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A088634 Index of the first occurrence of n in A066888.

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A191226 First occurrence of number n of Ramanujan primes in A191225.

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A191227 Last known occurrence of number n of Ramanujan primes in A191225.

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A230147 Record values in A165959.

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