A066888 -id:A066888 - cp's OEIS frontend

A088634 Index of the first occurrence of n in A066888.

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

A083419 Positions of records in A066888.

Original entry on oeis.org

0, 1, 8, 16, 27, 29, 35, 40, 50, 64, 91, 99, 107, 153, 157, 160, 171, 174, 196, 214, 235, 290, 320, 329, 333, 393, 444, 451, 480, 525, 545, 560, 564, 593, 662, 667, 708, 713, 730, 761, 791, 805, 865, 926, 1015, 1016, 1035, 1123, 1148, 1165, 1186, 1264, 1366

Offset: 1

Jason Earls, Jun 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A066888.

m = 1500; v = PrimePi[#[[2]]] - PrimePi[#[[1]]] & /@ Partition[Accumulate[Range[0, m + 1]], 2, 1]; seq = {}; vm = -1; Do[If[v[[k + 1]] > vm, vm = v[[k + 1]]; AppendTo[seq, k]], {k, 0, m}]; seq (* Amiram Eldar, Dec 14 2019 after Harvey P. Dale at A066888 *)

Data corrected by Amiram Eldar, Dec 14 2019

A083382 Write the numbers from 1 to n^2 consecutively in n rows of length n; a(n) = minimal number of primes in a row.

Original entry on oeis.org

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

Offset: 1

James Propp, Jun 05 2003

Conjectured by Schinzel (Hypothesis H2) to be always positive for n > 1.
The conjecture has been verified for n = prime < 790000 by Aguilar.
If this is true, then Legendre's conjecture is true as well. (See A014085). - Antti Karttunen, Jan 01 2019

			For n = 3 the array is
1 2 3 (2 primes)
4 5 6 (1 prime)
7 8 9 (1 prime)
so a(3) = 1

P. Ribenboim, The New Book of Prime Number Records, Chapter 6.
P. Ribenboim, The Little Book Of Big Primes, Springer-Verlag, NY 1991, page 185.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Carlos Rivera, The calendar-like square conjecture
A. Schinzel and W. Sierpinski, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.

A084927 generalizes this to three dimensions.
Cf. A083415, A083383, A066888, A092556, A092557. See A083414 for primes in columns.
Cf. A139326.
Cf. A000720, A010051, A014085.

a083382 n = f n n a010051_list where
   f m 0 _     = m
   f m k chips = f (min m $ sum chin) (k - 1) chips' where
     (chin,chips') = splitAt n chips
-- Reinhard Zumkeller, Jun 10 2012

A083382 := proc(n) local t1,t2,at; t1 := n; at := 0; for i from 1 to n do t2 := 0; for j from 1 to n do at := at+1; if isprime(at) then t2 := t2+1; fi; od; if t2 < t1 then t1 := t2; fi; od; t1; end;

Table[minP=n; Do[s=0; Do[If[PrimeQ[c+(r-1)*n], s++ ], {c, n}]; minP=Min[s, minP], {r, n}]; minP, {n, 100}]
Table[Min[Count[#,?PrimeQ]&/@Partition[Range[n^2],n]],{n,110}] (* _Harvey P. Dale, May 29 2013 *)

A083382(n) = { my(m=-1); for(i=0,n-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

Edited by Charles R Greathouse IV, Jul 07 2010

A190661 a(n) is the least number m such that there are at least n primes in the range (T(k-1), T(k)] for all k >= m, where T(k) is the k-th triangular number.

Original entry on oeis.org

1, 7, 16, 33, 52, 66, 79, 72, 109, 93, 121, 119, 130, 153, 169, 194, 180, 222, 235, 275, 294, 267, 256, 296, 329, 339, 333, 420, 383, 373, 372, 454, 396, 443, 449, 504, 463, 574, 559, 512, 592, 602, 596, 541, 652, 585, 683, 656, 687, 689, 708

Offset: 0

John W. Nicholson, May 18 2011

nonn

All values and even whether the sequence is well defined are conjectural.
a(n) is the conjectured index of the last occurrence of n in A066888.
It is conjectured that for every n >= 0, a(n) > n.
With R_n the n-th Ramanujan prime (A104272), it is conjectured that for every n >= 0, (1/2) R_n <= a(n) < (20/13) R_n. These bounds have been verified for all n up to 8000. For most n <= 8000, we have a(n) > R_n, with exceptions listed in A190881.

			Because it appears that A066888(7) = 1 is the last 1 of that sequence, a(1) = 7.

T. D. Noe, Table of n, a(n) for n = 0..8000

Cf. A066888, A000217, A000040, A088634, A104272, A190881.

Edited by T. D. Noe, May 19 2011

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

A057062 Let R(i,j) be the infinite square array with antidiagonals 1; 2,3; 4,5,6; ...; the n-th prime is in antidiagonal a(n).

Original entry on oeis.org

2, 2, 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

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

The smallest integer in the j-th antidiagonal is A000124(j-1). So a(n) is the index j such that A000124(j-1) <= prime(n) < A000124(j). - R. J. Mathar, Dec 02 2011

			The array begins
   1  3  6 10 15 ...
   2  5  9 14 ...
   4  8 13 ...
   7 12 ...
  11 ...
  ...
The third prime, 5, is in the 3rd antidiagonal, so a(3) = 3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A057045, A057048, A022846, A057057, A057054. A066888 counts how many times each positive integer appears in this sequence.

Cf. A010051.

a057062 n = a057062_list !! (n-1)
a057062_list = f 1 [1..] where
   f j xs = (replicate (sum $ map a010051 dia) j) ++ f (j + 1) xs'
     where (dia, xs') = splitAt j xs
-- Reinhard Zumkeller, Jul 26 2012

Table[Round[Sqrt[2*Prime[n]]], {n, 100}] (* T. D. Noe, Dec 03 2011 *)

a(n)=(sqrtint(8*prime(n))+1)\2 \\ Charles R Greathouse IV, Jul 26 2012

from math import isqrt
from sympy import prime
def A057062(n): return isqrt(prime(n)<<3)+1>>1 # Chai Wah Wu, Jun 19 2024

a(n) = round(sqrt(2*prime(n))). - Vladeta Jovovic, Jun 14 2003

A220492 Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 4, 1, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 4, 3, 1, 4, 2, 4, 3, 3, 3, 4, 4, 3, 4, 3, 2, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 5, 4, 4, 4, 4, 5, 5, 5, 4, 6, 4, 4, 5, 5, 5, 7, 2, 3, 6, 6, 6, 6, 5, 8, 4, 5, 6, 5, 4, 7

Offset: 0

Omar E. Pol, Feb 04 2013

nonn

It appears that a(n) > 0, if n > 1.
Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013
For n > 0, also the number of primes per quarter revolution of the Ulam Spiral. The conjecture implies that there is at least one prime in every turn after the first. - Ruud H.G. van Tol, Jan 30 2024

			When the nonnegative integers are written as an irregular triangle in which the right border gives the quarter-squares without repetitions, a(n) is the number of primes in the n-th row of triangle. See below (note that the prime numbers are in parenthesis):
---------------------------------------
Triangle                          a(n)
---------------------------------------
0;                                 0
1;                                 0
(2);                               1
(3),   4;                          1
(5),   6;                          1
(7),   8,   9;                     1
10,  (11), 12;                     1
(13), 14,  15,   16;               1
(17), 18, (19),  20;               2
21,   22, (23),  24,  25;          1
26,   27,  28,  (29), 30;          1
...

Ruud H.G. van Tol, Table of n, a(n) for n = 0..10000
Wikipedia, Oppermann's conjecture

Partial sums give A220506.

Cf. A000040, A002620, A001477, A014085, A066888, A073882, A222030.

a(n) = #primes([n^2/4, (n+1)^2/4]); \\ Ruud H.G. van Tol, Feb 01 2024

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

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.

A083403 Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of squarefree integers in n-th row.

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 3, 6, 6, 5, 7, 8, 8, 8, 10, 8, 10, 11, 12, 13, 15, 11, 14, 14, 16, 14, 15, 19, 19, 19, 18, 20, 19, 21, 21, 22, 24, 21, 24, 26, 22, 27, 26, 26, 27, 27, 29, 30, 30, 31, 32, 32, 33, 32, 34, 35, 35, 33, 36, 34, 36, 37, 40, 37, 42, 40, 41, 43, 42, 44, 40, 45, 44, 46

Offset: 1

Jason Earls, Jun 07 2003

			Triangle begins
1 (1 squarefree)
2 3 (2 squarefree)
4 5 6 (2 squarefree)
7 8 9 10 (2 squarefree)
11 12 13 14 15 (4 squarefree)
16 17 18 19 20 21 (3 squarefree)

Cf. A066888, A005117.

{ts(m)=local(r, t); r=1; for(n=1,m,t=0; for(k=r,n+r-1,if(issquarefree(k),t++)); print1(t","); r=n+r;) }

cp's OEIS Frontend

A088634 Index of the first occurrence of n in A066888.

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A083419 Positions of records in A066888.

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A083382 Write the numbers from 1 to n^2 consecutively in n rows of length n; a(n) = minimal number of primes in a row.

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A190661 a(n) is the least number m such that there are at least n primes in the range (T(k-1), T(k)] for all k >= m, where T(k) is the k-th triangular number.

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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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A057062 Let R(i,j) be the infinite square array with antidiagonals 1; 2,3; 4,5,6; ...; the n-th prime is in antidiagonal a(n).

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A220492 Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n).

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