A250473 -id:A250473 - 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

A250474 Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^3: a(n) = A078898(A030078(n)).

Original entry on oeis.org

4, 5, 9, 14, 28, 36, 57, 67, 93, 139, 154, 210, 253, 272, 317, 396, 473, 504, 593, 658, 687, 792, 866, 979, 1141, 1229, 1270, 1356, 1397, 1496, 1849, 1947, 2111, 2159, 2457, 2514, 2695, 2880, 3007, 3204, 3398, 3473, 3828, 3904, 4047, 4121, 4583, 5061, 5228, 5309, 5474, 5743, 5832, 6269, 6543, 6816, 7107, 7197, 7488, 7686, 7784, 8295, 9029, 9248, 9354, 9568, 10351

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Position of the first composite number (which is always 4) on row n of A249821. The fourth column of A249822.
Position of the first nonfixed term on row n of arrays of permutations A251721 and A251722.
According to the definition, this is the number of multiples of prime(n) below prime(n)^3 (and thus, the number of numbers below prime(n)^2) which do not have a smaller factor than prime(n). That is, the numbers remaining below prime(n)^2 after deleting all multiples of primes less than prime(n), as is done by applying the first n-1 steps of the sieve of Eratosthenes (when the first step is elimination of multiples of 2). This explains that the first differences are a(n+1)-a(n) = A050216(n)-1 for n>1, and a(n) = A054272(n)+2. - M. F. Hasler, Dec 31 2014

			prime(1) = 2 occurs as the least prime factor in range [1,8] for four times (all even numbers <= 8), thus a(1) = 4.
prime(2) = 3 occurs as the least prime factor in range [1,27] for five times (when n is: 3, 9, 15, 21, 27), thus a(2) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..5001

One more than A250473. Two more than A054272.
Column 4 of A249822.
Cf. also A250477 (column 6), A250478 (column 8).
Cf. A000040, A000879, A001248, A002110, A005867, A008683, A008836, A020639, A030078, A055396, A078898, A249821, A251721, A251722.

f[n_] := Count[Range[Prime[n]^3], x_ /; Min[First /@ FactorInteger[x]] == Prime@ n]; Array[f, 16] (* Michael De Vlieger, Mar 30 2015 *)

A250474(n) = 3 + primepi(prime(n)^2) - n; \\ Fast implementation.
for(n=1, 5001, write("b250474.txt", n, " ", A250474(n)));
\\ The following program reflects the given sum formula, but is far from the optimal solution:
allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A020639(n) = if(1==n,n,vecmin(factor(n)[,1]));
A055396(n) = if(1==n,0,primepi(A020639(n)));
A250474(n) = { my(p2 = prime(n)^2); sumdiv(A002110(n-1), d, moebius(d)*(p2\d)); };
for(n=1, 23, print1(A250474(n),", "));

(define (A250474 n) (let loop ((k 2)) (if (not (prime? (A249821bi n k))) k (loop (+ k 1))))) ;; This is even slower. Code for A249821bi given in A249821.

a(n) = 3 + A000879(n) - n = A054272(n) + 2 = A250473(n) + 1.
a(n) = A078898(A030078(n)).
a(1) = 1, a(n) = Sum_{d|A002110(n-1)} moebius(d)*floor(prime(n)^2/d). [Follows when A030078(n), prime(n)^3 is substituted to the similar formula given for A078898(n). Here A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use also Liouville's lambda (A008836) instead of Moebius mu (A008683)].
Other identities. For all n >= 1:
A249821(n, a(n)) = 4.

A054272 Number of primes in the interval [prime(n), prime(n)^2].

Original entry on oeis.org

2, 3, 7, 12, 26, 34, 55, 65, 91, 137, 152, 208, 251, 270, 315, 394, 471, 502, 591, 656, 685, 790, 864, 977, 1139, 1227, 1268, 1354, 1395, 1494, 1847, 1945, 2109, 2157, 2455, 2512, 2693, 2878, 3005, 3202, 3396, 3471, 3826, 3902, 4045, 4119, 4581, 5059

Offset: 1

Labos Elemer, May 05 2000

nonn

These primes are candidates for fortunate numbers (A005235).
These are precisely the primes available for the solution of Aguilar's conjecture or Haga's conjecture in Carlos Rivera's The Prime Puzzles and Problems Connection, (conjecture 26). Aguilar's conjecture states that at least one prime will be available for placement on each row and column of a p X p square array. Haga's conjecture states that just p primes are required for such placement in any p X p array. - Enoch Haga, Jan 23 2002
Also number of times p_n (the n-th prime) occurs as the least prime factor (A020639) among numbers in range [(p_n)+1, ((p_n)^3)-1]. For n=1, p_1 = 2 and there are two even numbers in range [3, 7], namely 4 and 6, so a(1) = 2. See also A250474. - Antti Karttunen, Dec 05 2014
The number of consecutive primes after the leading 1 in the prime(n)-rough numbers. - Benedict W. J. Irwin, Mar 24 2016

			n=4, the zone in question is [7,49] and encloses a(4)=12 primes, as follows: {7,11,13,17,19,23,29,31,37,41,43,47}.

Antti Karttunen, Table of n, a(n) for n = 1..5000
Carlos Rivera, Conjecture 26. The Calendar-like square Conjecture, The Prime Puzzles and Problems Connection.

One less than A250473, two less than A250474.
First differences: A251723.
Cf. A000040, A000879, A001248, A002110, A005235, A005867, A008683, A008836, A020639, A078898, A249747.

a[n_] := PrimePi[Prime[n]^2] - n + 1; Array[a, 50] (* Jean-François Alcover, Dec 07 2015 *)

\\ A fast version:
default(primelimit, 2^31 + 2^30);
A054272(n) = 1 + primepi(prime(n)^2) - n;
for(n=1, 5000, write("b054272.txt", n, " ", A054272(n)));
\\ The following mirrors the given new formula. It is far from an optimal way to compute this sequence:
allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A054272(n) = { my(p2); p2 = prime(n)^2; sumdiv(A002110(n), d, moebius(d)*floor(p2/d)); };
for(n=1, 22, print1(A054272(n),", ")); \\ Antti Karttunen, Dec 05 2014

a(n) = A000879(n) - n + 1.
From Antti Karttunen, Dec 05-08 2014: (Start)
a(n) = A250473(n) - 1 = A250474(n) - 2.
a(n) = sum_{d | A002110(n)} moebius(d) * floor((p_n)^2 / d). [Where p_n is the n-th prime (A000040(n)) and A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could also use Liouville's lambda (A008836) instead of Moebius mu (A008683).]
The ratio (a(n) * A002110(n)) / (A001248(n) * A005867(n)) stays near 1, which follows from the above summation formula. See also A249747.
(End)

A251721 Square array of permutations: A(row,col) = A249822(row, A249821(row+1, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 4, 4, 3, 2, 1, 7, 6, 4, 3, 2, 1, 11, 7, 5, 4, 3, 2, 1, 6, 9, 6, 5, 4, 3, 2, 1, 13, 10, 7, 6, 5, 4, 3, 2, 1, 17, 5, 8, 7, 6, 5, 4, 3, 2, 1, 10, 12, 10, 8, 7, 6, 5, 4, 3, 2, 1, 19, 15, 11, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 13, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 8, 16, 14, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 19, 15, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Antti Karttunen, Dec 07 2014

These are the "first differences" between permutations of array A249821, in a sense that by composing the first k rows of this array [from left to right, as in a(n) = row_1(row_2(...(row_k(n))))], one obtains row k+1 of A249821.

On row n, the first A250473(n) terms are fixed, and the first non-fixed term comes at A250474(n).

			The top left corner of the array:
1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 10, 19, 9, 8, 23, 29, 14, 15, 31, 22, ...
1, 2, 3, 4, 6, 7, 9, 10, 5, 12, 15, 8, 16, 19, 21, 22, 13, 24, 11, 27, ...
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 9, 16, 18, 20, 21, 23, 24, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, ...
...

Inverse permutations can be found from array A251722.
Row 1: A064216, Row 2: A249745, Row 3: A250475.
Cf. A000027, A002260, A004736, A078898, A083221, A246277, A246278, A249821, A249822, A250473, A250474.

(define (A251721bi row col) (A249822bi row (A249821bi (+ row 1) col)))
(define (A251721 n) (A251721bi (A002260 n) (A004736 n)))
;; Code for A249821bi and A249822bi given in A249821, A249822.

A(row,col) = A249822(row, A249821(row+1, col)).

A(row,col) = A078898(A246278(row, A246277(A083221(row+1, col)))).

A251722 Square array of permutations: A(row,col) = A249822(row+1, A249821(row, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 4, 4, 3, 2, 1, 8, 9, 4, 3, 2, 1, 6, 5, 5, 4, 3, 2, 1, 14, 6, 6, 5, 4, 3, 2, 1, 13, 12, 7, 6, 5, 4, 3, 2, 1, 11, 7, 8, 7, 6, 5, 4, 3, 2, 1, 7, 8, 14, 8, 7, 6, 5, 4, 3, 2, 1, 23, 19, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 10, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 17, 21, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 42, 11, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Antti Karttunen, Dec 07 2014

These are the "first differences" between permutations of array A249822, in a sense that by composing the first k rows of this array [from right to left, as in a(n) = row_k(...(row_2(row_1(n))))], one obtains row k+1 of A249822.

On row n the first non-fixed term is A250474(n+1) at position A250474(n), i.e., on row 1 it is 5 at n=4, on row 2 it is 9 at n=5, on row 3 it is 14 at n=9, etc. All the previous A250473(n) terms are fixed.

			The top left corner of the array:
1, 2, 3, 5, 4, 8, 6, 14, 13, 11, 7, 23, 9, 17, 18, 41, 10, 38, 12, 32, ...
1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, ...
1, 2, 3, 4, 5, 6, 7, 8, 14, 9, 10, 21, 11, 12, 13, 15, 33, 16, 25, 17, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 14, 15, 16, 17, 18, 19, ...
...

Inverse permutations can be found from array A251721.
Row 1: A048673, Row 2: A249746, Row 3: A250476.
Cf. A000027, A002260, A004736, A078898, A083221, A246277, A246278, A249821, A249822, A250473, A250474.

(define (A251722bi row col) (A249822bi (+ row 1) (A249821bi row col)))
(define (A251722 n) (A251722bi (A002260 n) (A004736 n)))
;; Code for A249821bi and A249822bi given in A249821, A249822.

A(row,col) = A249822(row+1, A249821(row, col)).

A(row,col) = A078898(A246278(row+1, A246277(A083221(row, col)))).

cp's OEIS Frontend

A251723 First differences of A054272, A250473 and A250474: a(n) = A054272(n+1) - A054272(n).

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

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A250474 Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^3: a(n) = A078898(A030078(n)).

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A054272 Number of primes in the interval [prime(n), prime(n)^2].

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A251721 Square array of permutations: A(row,col) = A249822(row, A249821(row+1, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A251722 Square array of permutations: A(row,col) = A249822(row+1, A249821(row, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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