A256470 -id:A256470 - cp's OEIS frontend

A050216 Number of primes between (prime(n))^2 and (prime(n+1))^2, with a(0) = 2 by convention.

2, 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, 57, 44, 20, 46, 80, 78, 32, 90, 66, 30, 106, 75, 114, 163, 89, 42, 87, 42, 100, 354, 99, 165, 49, 299, 58, 182, 186, 128, 198, 195, 76, 356, 77, 144, 75, 463, 479, 168, 82, 166, 270, 90, 438, 275, 274, 292, 91, 292, 199, 99

Offset: 0

Eric W. Weisstein

The function in Brocard's Conjecture, which states that for n >= 2, a(n) >= 4.
The lines in the graph correspond to prime gaps of 2, 4, 6, ... . - T. D. Noe, Feb 04 2008
Lengths of blocks of consecutive primes in A000430 (union of primes and squares of primes). - Reinhard Zumkeller, Sep 23 2011
In the n-th step of the sieve of Eratosthenes, all multiples of prime(n) are removed. Then a(n) gives the number of new primes obtained after the n-th step. - Jean-Christophe Hervé, Oct 27 2013
More precisely, after the n-th step, one is sure to have eliminated all composites less than prime(n+1)^2, since any composite N has a prime factor <= sqrt(N). It is in exactly this (restricted) sense that a(n) yields the number of "new primes" (additional numbers known to be prime) after the n-th step. But one knows after the n-th step also that all remaining numbers between prime(n+1)^2 and prime(n+1)*(prime(n+1)+2) are prime: By construction they don't have a factor less than prime(n+1) and they don't have a factor prime(n+1) so the least prime factor could be prime(n+2) >= prime(n+1)+2. For example, after eliminating multiples of 3 in the 2nd step, one has (2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 31, 35, ...) and one knows that all remaining numbers strictly in between 5^2=25 and 5*(5+2)=35 are prime, too. - M. F. Hasler, Dec 31 2014
Numerically, the slope of the lowest "ray" m(n) = min {a(k); k>n}, seems to converge to a value somewhere in the range 1.75 < m(n)/n < 1.8; with m(n)/n > 1.7 for n > 900, m(n)/n > 1.75 for n > 2700. - M. F. Hasler, Dec 31 2014
Legendre's conjecture (see A014085) would imply that a(n) >= 2 for all n and that sequences A054272, A250473 and A250474 were thus strictly increasing (see the Wikipedia article about Brocard's conjecture). - Antti Karttunen, Jan 01 2015
a(n) >= 4 up to at least n = 4*10^5. - Eric W. Weisstein, Jan 13 2025

			There are 2 primes less than 2^2, there are 2 primes between 2^2 and 3^2, 5 primes between 3^2 and 5^2, etc. [corrected by Jonathan Sperry, Aug 30 2013]

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

T. D. Noe, Table of n, a(n) for n = 0..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 16.
Anthony G. Shannon and Jean V. Leyendekkers, On Legendre's Conjecture, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017): 117-125.
Nicolas Vaillant, Graph of A050216(n) / (n * A001223(n))
Nicolas Vaillant, Average density of primes between prime(n)^2 and prime(n+1)^2
Eric Weisstein's World of Mathematics, Brocard's Conjecture.
Wikipedia, Brocard's Conjecture

First differences of A000879.
One more than A251723.
Cf. A010051, A001248, A014085, A089609, A251719, A256468, A256469, A256470, A029707, A137859.
Cf. A380135 (High water marks for number of primes between prime(n)^2 and prime(n+1)^2).
Cf. A380136 (Positions of the high water marks for number of primes between prime(n)^2 and prime(n+1)^2).

import Data.List (group)
a050216 n = a050216_list !! (n-1)
a050216_list =
   map length $ filter (/= [0]) $ group $ map a010051 a000430_list
-- Reinhard Zumkeller, Sep 23 2011

A050216 := proc(n)
    local p,pn ;
    if n = 0 then
        2;
    else
        p := ithprime(n) ;
        pn := nextprime(p) ;
        numtheory[pi](pn^2)-numtheory[pi](p^2) ;
    end if;
end proc:
seq(A050216(n),n=0..40) ; # R. J. Mathar, Jan 27 2025

-Subtract @@@ Partition[PrimePi[Prime[Range[20]]^2], 2, 1] (* Eric W. Weisstein, Jan 10 2025 *)

a(n)={n||return(2);primepi(prime(n+1)^2)-primepi(prime(n)^2)} \\ M. F. Hasler, Dec 31 2014

For all n >= 1, a(n) = A256468(n) + A256469(n). - Antti Karttunen, Mar 30 2015

Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 1. - Alain Rocchelli, Sep 30 2023

Edited by N. J. A. Sloane, Nov 15 2009

A256471 Numbers n for which there is an equal number of primes in range [prime(n)^2, prime(n)prime(n+1)] as there are primes in range [prime(n)prime(n+1), prime(n+1)^2].

Original entry on oeis.org

1, 10, 14, 17, 20, 90, 110, 152, 176, 185, 193, 230, 344, 377, 391, 392, 404, 441, 442, 542, 1066, 1533, 1550, 1632, 1638, 1639, 1810, 2115, 2210, 2302, 2567, 2768, 2921, 3172, 3518, 3615, 3764, 4357, 4577, 4787, 4853, 5060, 5278, 6329

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions of zeros in A256470.

Cf. A256472, A256473 (the corresponding primes).

A256468 Number of primes between prime(n)^2 and prime(n)*prime(n+1).

Original entry on oeis.org

1, 2, 2, 6, 4, 8, 5, 12, 22, 8, 27, 21, 11, 23, 38, 36, 16, 43, 31, 15, 52, 36, 52, 75, 45, 22, 42, 19, 48, 160, 47, 81, 22, 141, 26, 90, 89, 65, 102, 96, 40, 180, 40, 73, 38, 227, 227, 85, 44, 85, 129, 43, 216, 133, 140, 137, 45, 147, 105, 46, 260, 354, 115, 52, 108, 386, 165, 283, 64

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6542
A. Karttunen, Ratio a(n)/A256469(n) plotted with OEIS Plot2-script
A. Karttunen, Ratio a(n)/A050216(n) plotted with OEIS Plot2-script

One less than A256447.

Cf. A050216, A256469, A256470.

Table[Abs@ Subtract[PrimePi[Prime[n]^2], PrimePi[Prime[n] Prime[n + 1]]], {n, 69}] (* Michael De Vlieger, Mar 30 2015 *)
PrimePi[Times@@#]-PrimePi[#[[1]]^2]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Mar 31 2025 *)

allocatemem(234567890);
default(primelimit,4294965247);
A256468(n) = (primepi(prime(n)*prime(n+1)) - primepi(prime(n)^2));
for(n=1, 6542, write("b256468.txt", n, " ", A256468(n)));

(define (A256468 n) (let* ((p (A000040 n)) (p2 (* p p))) (let loop ((s 0) (k (* p (A000040 (+ 1 n))))) (cond ((= k p2) s) (else (loop (+ s (if (prime? k) 1 0)) (- k 1)))))))

a(n) = A256447(n)-1.
a(n) = A050216(n) - A256469(n).
a(n) = A256469(n) - A256470(n).

A256449 a(n) = A256447(n) - A256448(n).

Original entry on oeis.org

3, 2, 1, 0, 2, -3, 2, 0, 0, 3, 0, 1, 5, 3, -1, -3, 3, -1, -1, 3, 1, 0, -7, -10, 4, 5, 0, -1, -1, -31, -2, 0, -2, -14, -3, 1, -5, 5, 9, 0, 7, 7, 6, 5, 4, -6, -22, 5, 9, 7, -9, -1, -3, -6, 9, -15, 2, 5, 14, -4, 11, -24, 13, 0, 4, -9, -8, -10, 6, -2, 0, -2, 16, 11, -7, -13, 7, -11, 0

Offset: 1

Antti Karttunen, Mar 29 2015

sign

Positions of zeros: 4, 8, 9, 11, 22, 27, 32, 40, 64, 71, 79, 104, 113, 126, 140, 201, 225, 332, 333, 394, 451, ...

Corresponding primes: 7, 19, 23, 31, 79, 103, 131, 173, 311, 353, 401, 569, 617, 701, 809, 1229, 1427, 2237, 2239, 2707, 3187, ...

Antti Karttunen, Table of n, a(n) for n = 1..564
A. Karttunen, Sequences A256449 and A251723 compared with OEIS Plot2-script (See also the ratio-plots linked in A256447.)

Cf. A250474, A250477, A251723, A256446, A256447, A256448, A256470.

(define (A256449 n) (- (A256447 n) (A256448 n)))
(define (A256449 n) (- (* 2 (A250477 n)) (A250474 n) (A250474 (+ n 1))))

a(n) = A256447(n) - A256448(n).
a(n) = 2*A250477(n) - A250474(n) - A250474(n+1).
a(n) = 3 - A256470(n).

A256469 Number of primes between prime(n)*prime(n+1) and prime(n+1)^2.

Original entry on oeis.org

1, 3, 4, 9, 5, 14, 6, 15, 25, 8, 30, 23, 9, 23, 42, 42, 16, 47, 35, 15, 54, 39, 62, 88, 44, 20, 45, 23, 52, 194, 52, 84, 27, 158, 32, 92, 97, 63, 96, 99, 36, 176, 37, 71, 37, 236, 252, 83, 38, 81, 141, 47, 222, 142, 134, 155, 46, 145, 94, 53, 252, 381, 105, 55, 107, 398, 176, 296, 61

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

			For n=1, there is only one prime in range prime(1)*prime(2) .. prime(2)^2, [6 .. 9], namely 7, thus a(1) = 1.
For n=2, the primes in range prime(2)*prime(3) .. prime(3)^2, [15 .. 25] are {17, 19, 23}, thus a(2) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..6541

Cf. A050216, A256448, A256468, A256470.

Table[Count[Range[Prime[n] Prime[n + 1], Prime[n + 1]^2], ?PrimeQ], {n, 69}] (* _Michael De Vlieger, Mar 30 2015 *)
Table[PrimePi[Prime[n+1]^2]-PrimePi[Prime[n]Prime[n+1]],{n,70}] (* Harvey P. Dale, Jul 31 2021 *)

allocatemem(234567890);
default(primelimit,4294965247);
A256469(n) = (primepi(prime(n+1)^2) - primepi(prime(n)*prime(n+1)));
for(n=1, 6541, write("b256469.txt", n, " ", A256469(n)));

(define (A256469 n) (let* ((p (A000040 n)) (q (A000040 (+ 1 n))) (q2 (* q q))) (let loop ((s 0) (k (* p q))) (cond ((= k q2) s) (else (loop (+ s (if (prime? k) 1 0)) (+ k 1)))))))

a(n) = A256448(n)+2.
a(n) = A050216(n) - A256468(n).
a(n) = A256468(n) + A256470(n).

A256474 Numbers n for which there are at least as many primes in the range [prime(n)prime(n+1), prime(n+1)^2] as in the range [prime(n)^2, prime(n)prime(n+1)].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 46, 47, 51, 52, 53, 54, 56, 57, 60, 62, 64, 66, 67, 68, 70, 71, 72, 75, 76, 78, 79, 80, 82, 83, 84, 85, 87, 89, 90, 91, 93, 94, 95, 97, 99, 100

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions of nonnegative terms in A256470.

Antti Karttunen, Table of n, a(n) for n = 1..3457

Complement: A256475.
Union of A256471 and A256476.
Cf. A256484 (corresponding primes).
Cf. A256470.

Select[Range@100, Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] >= Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

A256475 Numbers n for which there are more primes in range [prime(n)^2, prime(n)prime(n+1)] than in range [prime(n)prime(n+1), prime(n+1)^2].

Original entry on oeis.org

13, 25, 26, 38, 39, 41, 42, 43, 44, 45, 48, 49, 50, 55, 58, 59, 61, 63, 65, 69, 73, 74, 77, 81, 86, 88, 92, 96, 98, 101, 103, 106, 107, 108, 109, 116, 117, 120, 121, 122, 124, 125, 128, 141, 142, 143, 145, 146, 148, 149, 151, 155, 158, 159, 166, 169, 172, 173, 177, 179, 181, 182, 183, 190, 191, 194, 195, 196, 197, 206

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions of negative terms in A256470.

Equally: Numbers n for which there are less primes in range [prime(n)*prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)*prime(n+1)].

Antti Karttunen, Table of n, a(n) for n = 1..3084

Complement: A256474.
Setwise difference of A256477 and A256471.
Cf. A256485 (corresponding primes).
Cf. A256470.

Select[Range@210, Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] > Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

A256476 Numbers n for which there are more primes in range [prime(n)prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)prime(n+1)].

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 46, 47, 51, 52, 53, 54, 56, 57, 60, 62, 64, 66, 67, 68, 70, 71, 72, 75, 76, 78, 79, 80, 82, 83, 84, 85, 87, 89, 91, 93, 94, 95, 97, 99, 100, 102, 104, 105, 111, 112, 113, 114, 115, 118, 119, 123

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions where A256470 is strictly positive.

Antti Karttunen, Table of n, a(n) for n = 1..3413

Complement: A256477.
Setwise difference of A256474 and A256471.
Cf. A256470.

Select[Range@125, Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] > Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

A256477 Numbers n for which the number of primes in the range [prime(n)prime(n+1), prime(n+1)^2] is less than or equal to the number of primes in the range [prime(n)^2, prime(n)prime(n+1)].

Original entry on oeis.org

1, 10, 13, 14, 17, 20, 25, 26, 38, 39, 41, 42, 43, 44, 45, 48, 49, 50, 55, 58, 59, 61, 63, 65, 69, 73, 74, 77, 81, 86, 88, 90, 92, 96, 98, 101, 103, 106, 107, 108, 109, 110, 116, 117, 120, 121, 122, 124, 125, 128, 141, 142, 143, 145, 146, 148, 149, 151, 152, 155, 158, 159, 166, 169

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions where A256470 is zero or negative.

Antti Karttunen, Table of n, a(n) for n = 1..3128

Complement: A256476.
Union of A256471 and A256475.
Cf. A256470.

Select[Range@ 170, Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] <= Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

cp's OEIS Frontend

A050216 Number of primes between (prime(n))^2 and (prime(n+1))^2, with a(0) = 2 by convention.

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A256471 Numbers n for which there is an equal number of primes in range [prime(n)^2, prime(n)*prime(n+1)] as there are primes in range [prime(n)*prime(n+1), prime(n+1)^2].

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A256468 Number of primes between prime(n)^2 and prime(n)*prime(n+1).

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A256449 a(n) = A256447(n) - A256448(n).

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A256469 Number of primes between prime(n)*prime(n+1) and prime(n+1)^2.

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A256474 Numbers n for which there are at least as many primes in the range [prime(n)*prime(n+1), prime(n+1)^2] as in the range [prime(n)^2, prime(n)*prime(n+1)].

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A256475 Numbers n for which there are more primes in range [prime(n)^2, prime(n)*prime(n+1)] than in range [prime(n)*prime(n+1), prime(n+1)^2].

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A256476 Numbers n for which there are more primes in range [prime(n)*prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)*prime(n+1)].

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A256471 Numbers n for which there is an equal number of primes in range [prime(n)^2, prime(n)prime(n+1)] as there are primes in range [prime(n)prime(n+1), prime(n+1)^2].

A256474 Numbers n for which there are at least as many primes in the range [prime(n)prime(n+1), prime(n+1)^2] as in the range [prime(n)^2, prime(n)prime(n+1)].

A256475 Numbers n for which there are more primes in range [prime(n)^2, prime(n)prime(n+1)] than in range [prime(n)prime(n+1), prime(n+1)^2].

A256476 Numbers n for which there are more primes in range [prime(n)prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)prime(n+1)].

A256477 Numbers n for which the number of primes in the range [prime(n)prime(n+1), prime(n+1)^2] is less than or equal to the number of primes in the range [prime(n)^2, prime(n)prime(n+1)].