A083382 -id:A083382 - cp's OEIS frontend

A083383 Positions of records in A083382.

1, 2, 9, 26, 42, 57, 75, 76, 103, 116, 122, 143, 151, 172, 191, 197, 224, 236, 251, 266, 288, 316, 327, 338, 356, 372, 385, 401, 451, 482, 490, 501, 541, 558, 578, 586, 621, 636, 644, 670, 678, 692, 724, 735, 776, 801, 826, 851, 864, 872, 890, 906, 915, 924

Offset: 0

N. J. A. Sloane, Jun 06 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

a083383 n = a083383_list !! (n-1)
a083383_list = 1 : f 0 [2..] (tail a083382_list) where
   f m (x:xs) (y:ys) | y <= m    = f m xs ys
                     | otherwise = x : f y xs ys
-- Reinhard Zumkeller, Jun 10 2012

More terms from Vladeta Jovovic, Jun 06 2003

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

A083414 Write the numbers from 1 to n^2 consecutively in n rows of length n; let c(k) = number of primes in k-th column; a(n) = minimal c(k) for gcd(k,n) = 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 2, 3, 5, 2, 6, 1, 5, 5, 5, 2, 10, 2, 6, 5, 8, 3, 9, 5, 8, 5, 9, 4, 17, 3, 9, 7, 9, 6, 15, 4, 9, 8, 13, 4, 21, 3, 11, 10, 11, 4, 17, 5, 15, 9, 14, 5, 20, 8, 14, 9, 14, 6, 27, 6, 15, 12, 14, 9, 26, 6, 15, 12, 23, 5, 25, 3, 15, 13, 17, 8, 29, 7, 20, 12, 17, 7, 32

Offset: 1

N. J. A. Sloane, Jun 10 2003

nonn

Conjectured to be always positive for n>1.
Note that a(n) is large when phi(n), the number of integers relatively prime to n, is small and vice versa. - T. D. Noe, Jun 10 2003
The conjecture is true for all n <= 40000.

			For n = 4 the array is
.   1  2  3  4
.   5  6  7  8
.   9 10 11 12
.  13 14 15 16
in which columns 1 and 3 contain 2 and 3 primes; therefore a(4) = 2.

See A083382 for references and links.

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A083415 and A083382 for primes in rows.
A084927 generalizes this to three dimensions.
Cf. A010051.

a083414 n = minimum $ map c $ filter ((== 1) . (gcd n)) [1..n] where
   c k = sum $ map a010051 $ enumFromThenTo k (k + n) (n ^ 2)
-- Reinhard Zumkeller, Jun 10 2012

Table[minP=n; Do[If[GCD[c, n]==1, s=0; Do[If[PrimeQ[c+(r-1)*n], s++ ], {r, n}]; minP=Min[s, minP]], {c, n}]; minP, {n, 100}]

More terms from Vladeta Jovovic and T. D. Noe, Jun 10 2003

A083415 Triangle read by rows: T(n,k) is defined as follows. Write the numbers from 1 to n^2 consecutively in n rows of length n; T(n,k) = number of primes in k-th row.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, following a suggestion of Wouter Meeussen, Jun 10 2003

Sum(T(n,k): 1<=k<=n) = A038107(n); T(n,1)=A000720(n); T(n,2)=A060715(n) for n>1. - Reinhard Zumkeller, Jan 07 2004

			{0}
{1, 1}
{2, 1, 1} from / 1 2 3 / 4 5 6 / 7 8 9 /
{2, 2, 1, 1}
{3, 1, 2, 2, 1}
{3, 2, 2, 2, 1, 1}

Paulo Ribenboim, "The Little Book Of Big Primes," Springer-Verlag, NY 1991, page 185.

T. D. Noe, Rows n=1..100 of triangle, flattened

Cf. A083382, A083414, A092556, A092557.
Cf. A139325.
Cf. A010051.

a083415 n k = a083415_row n !! (k-1)
a083415_row n = f n a010051_list where
   f 0 _     = []
   f k chips = (sum chin) : f (k - 1) chips' where
     (chin,chips') = splitAt n chips
a083415_tabl = map a083415_row [1..]
-- Reinhard Zumkeller, Jun 10 2012

Table[PrimePi[m n]-PrimePi[(m-1) n], {n, 17}, {m, n}]

A084927 If the numbers 1 to n^3 are arranged in a cubic array, a(n) is the minimum number of primes in each column of the n^2 columns in the "top view" that can have primes.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 0, 3, 1, 1, 0, 4, 0, 3, 1, 3, 0, 8, 0, 2, 2, 3, 1, 5, 0, 2, 1, 4, 0, 9, 0, 3, 2, 4, 0, 6, 1, 6, 2, 4, 0, 5, 0, 5, 2, 3, 0, 11, 0, 4, 3, 3, 1, 10, 1, 5, 3, 7, 0, 10, 0, 2, 4, 6, 2, 11, 1, 7, 3, 5, 0, 13, 2, 6, 4, 7, 1, 17, 2, 6, 2, 6, 2, 12, 1, 8, 4, 8

Offset: 1

T. D. Noe, Jun 12 2003

nonn

This is a three-dimensional generalization of A083414.

			For the case n=3, the numbers are arranged in a cubic array as follows:
1..2..3........10.11.12........19.20.21
4..5..6........13.14.15........22.23.24
7..8..9........16.17.18........25.26.27
The first column is (1,10,19), the second is (2,11,20), etc. Only columns whose tops are relatively prime to n are counted. In this case, columns starting with 3, 6 and 9 cannot have primes. a(n) = 0 for n = 1, 25, 55 and the primes from 5 to 83, except 67 and 79. It appears that a(n) > 0 for n > 83. This has been confirmed up to n = 1000.

See A083382 for references and links to the two-dimensional case.

Cf. A083382, A083414, A084928 (east-west view), A084929 (north-south view).

Table[minP=n; Do[If[GCD[c, n]==1, s=0; Do[If[PrimeQ[c+(r-1)*n^2], s++ ], {r, n}]; minP=Min[s, minP]], {c, n^2}]; minP, {n, 100}]

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)).

A084928 If the numbers 1 to n^3 are arranged in a cubic array, a(n) is the minimum number of primes in each row of the n^2 rows in the "east-west view" that can have primes.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2

Offset: 1

T. D. Noe, Jun 12 2003

nonn

This is a three-dimensional generalization of A083382.

			For the case n=3, the numbers are arranged in a cubic array as follows:
1..2..3........10.11.12........19.20.21
4..5..6........13.14.15........22.23.24
7..8..9........16.17.18........25.26.27
The first row is (1,2,3), the second is (4,5,6), etc. Surprisingly, a(n) = 0 for all n from 3 to 66. It appears that a(n) > 0 for n > 128. This has been confirmed up to n = 1000.

See A083382 for references and links to the two-dimensional case.

Antti Karttunen, Table of n, a(n) for n = 1..900

Cf. A083382, A083414, A084927 (top view), A084929 (north-south view).

Table[minP=n; Do[s=0; Do[If[PrimeQ[n*(c-1)+r], s++ ], {r, n}]; minP=Min[s, minP], {c, n^2}]; minP, {n, 100}]

A084928(n) = { my(m=-1); for(i=0,(n^2)-1,my(s=sum(j=(i*n),((i+1)*n)-1,isprime(1+j))); if((m<0) || (s < m), m = s)); (m); }; \\ Antti Karttunen, Jan 01 2019

More terms from Antti Karttunen, Jan 01 2019

A139326 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = minimal number of primes in a row.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 3, 3, 4, 5, 4, 6, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 6, 6, 5, 5, 5, 8, 6, 7, 7, 8, 7, 7, 8, 8, 8, 9, 9, 8, 9, 9, 10, 10, 9, 9, 8, 9, 9, 9, 10, 10, 11, 9, 11, 10, 11, 10, 11, 11, 11, 12, 12, 12, 13, 12, 13, 13, 13, 12, 13, 14, 15, 14, 12, 13, 13, 15, 14

Offset: 1

Reinhard Zumkeller, Apr 14 2008

nonn

a(n) = Min{A139325(n,k): 1<=k<=n}.

			a(4)=Min{#{3,5,7},#{11,13},#{17,19,23},#{29,31}}=Min{3,2,3,2}=2:
..1 ...3 ...5 ...7 ... primes in first row = {3,5,7},
..9 ..11 ..13 ..15 ... primes in 2nd row = {11,13},
.17 ..19 ..21 ..23 ... primes in 3rd row = {17,19},
.25 ..27 ..29 ..31 ... primes in 4th row = {29,31}.

Cf. A139327, A083382.

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.

cp's OEIS Frontend

A083383 Positions of records in A083382.

Views

Author

Keywords

Links

Programs

Haskell

Extensions

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

A083414 Write the numbers from 1 to n^2 consecutively in n rows of length n; let c(k) = number of primes in k-th column; a(n) = minimal c(k) for gcd(k,n) = 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A083415 Triangle read by rows: T(n,k) is defined as follows. Write the numbers from 1 to n^2 consecutively in n rows of length n; T(n,k) = number of primes in k-th row.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

A084927 If the numbers 1 to n^3 are arranged in a cubic array, a(n) is the minimum number of primes in each column of the n^2 columns in the "top view" that can have primes.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Perl

Formula

A084928 If the numbers 1 to n^3 are arranged in a cubic array, a(n) is the minimum number of primes in each row of the n^2 rows in the "east-west view" that can have primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs