A145445 -id:A145445 - cp's OEIS frontend

A145449 Numbers n such that A145445(n) = A145446(n).

16, 17, 18, 123, 124, 125, 126, 127, 128, 129, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 1797, 1798, 1799, 1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809, 1810, 1811, 1812, 1813, 1814, 1815, 1816, 1817, 1818

Offset: 1

Zak Seidov, Oct 10 2008

nonn

Numbers n such that s(n), the smallest square > n-th prime, equals q(n), the smallest cube > n-th prime, s(n) = A145445(n), q(n) = A145446(n).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A104103, A104147, A145445, A145446, A145447

Do[If[Ceiling[Prime[n]^(1/3)]^3 == Ceiling[Prime[n]^(1/2)]^2,Print[n]],{n,10000}]

A014085 Number of primes between n^2 and (n+1)^2.

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, 8, 7, 8, 9, 8, 8, 10, 9, 10, 9, 10, 9, 9, 12, 11, 12, 11, 9, 12, 11, 13, 10, 13, 15, 10, 11, 15, 16, 12, 13, 11, 12, 17, 13, 16, 16, 13, 17, 15, 14, 16, 15, 15, 17, 13, 21, 15, 15, 17, 17, 18, 22, 14, 18, 23, 13

Offset: 0

Jon Wild, Jul 14 1997

Suggested by Legendre's conjecture (still open) that for n > 0 there is always a prime between n^2 and (n+1)^2.
a(n) is the number of occurrences of n in A000006. - Philippe Deléham, Dec 17 2003
See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
Legendre's conjecture may be written pi((n+1)^2) - pi(n^2) > 0 for all positive n, where pi(n) = A000720(n), [the prime counting function]. - Jonathan Vos Post, Jul 30 2008 [Comment corrected by Jonathan Sondow, Aug 15 2008]
Legendre's conjecture can be generalized as follows: for all integers n > 0 and all real numbers k > K, there is a prime in the range n^k to (n+1)^k. The constant K is conjectured to be log(127)/log(16). See A143935. - T. D. Noe, Sep 05 2008
For n > 0: number of occurrences of n^2 in A145445. - Reinhard Zumkeller, Jul 25 2014

			a(17) = 5 because between 17^2 and 18^2, i.e., 289 and 324, there are 5 primes (which are 293, 307, 311, 313, 317).

J. R. Goldman, The Queen of Mathematics, 1998, p. 82.

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. 8.
Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation 68: (1999), 411-415.
Tsutomu Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
Mehdi Hassani, Counting primes in the interval (n^2, (n+1)^2), arXiv:math/0607096 [math.NT], 2006.
Edmund Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228.
Peter Munn, Logarithmic plot: number of primes between consecutive squares vs number of integers between the same squares
Michael Penn, Legendre's Conjecture is probably true, and here's why, YouTube video, 2023.
Hugo Pfoertner, One million terms in b-file format, 13.1 MB (2024).
Hugo Pfoertner, Plot of 10^6 sequence terms with conjectured scatter band, large pdf (13.6 MB) (2024).
Hugo Pfoertner, Lower limit of the scatter band represented as a step function.
Jonathan Sorenson and Jonathan Webster, An algorithm to verify Legendre’s conjecture up to 7*10^13, Research in Number Theory 11:4 (2025).
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

First differences of A038107.
Cf. A000006, A053000, A053001, A007491, A077766, A077767, A108954, A000720, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A010051, A061265, A221056, A000290, A145445.
Counts of primes between consecutive higher powers: A060199, A061235, A062517.
Cf. A333846, A349996, A349997, A349998, A349999.

a014085 n = sum $ map a010051 [n^2..(n+1)^2]
-- Reinhard Zumkeller, Mar 18 2012

Table[PrimePi[(n + 1)^2] - PrimePi[n^2], {n, 0, 80}] (* Lei Zhou, Dec 01 2005 *)
Differences[PrimePi[Range[0,90]^2]] (* Harvey P. Dale, Nov 25 2015 *)

a(n)=primepi((n+1)^2)-primepi(n^2) \\ Charles R Greathouse IV, Jun 15 2011

from sympy import primepi
def a(n): return primepi((n+1)**2) - primepi(n**2)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 05 2021

a(n) = A000720((n+1)^2) - A000720(n^2). - Jonathan Vos Post, Jul 30 2008
a(n) = Sum_{k = n^2..(n+1)^2} A010051(k). - Reinhard Zumkeller, Mar 18 2012
Conjecture: for all n>1, abs(a(n)-(n/log(n))) < sqrt(n). - Alain Rocchelli, Sep 20 2023
Up to n=10^6 there are no counterexamples to this conjecture. - Hugo Pfoertner, Dec 16 2024
Sorenson & Webster show that a(n) > 0 for all 0 < n < 7.05 * 10^13. - Charles R Greathouse IV, Jan 31 2025

A245508 Smallest double square (cf. A001105) greater than n-th prime.

Original entry on oeis.org

2, 8, 8, 8, 18, 18, 18, 32, 32, 32, 32, 50, 50, 50, 50, 72, 72, 72, 72, 72, 98, 98, 98, 98, 98, 128, 128, 128, 128, 128, 128, 162, 162, 162, 162, 162, 162, 200, 200, 200, 200, 200, 200, 200, 200, 200, 242, 242, 242, 242, 242, 242, 242, 288, 288, 288, 288

Offset: 1

Reinhard Zumkeller, Jul 25 2014

nonn

For n > 2: prime(n) < a(n) < 2*prime(n) and a(n) = A245499(A000040(n),2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A001105, A145445.

import Data.List (genericIndex)
a245508 n = genericIndex a245508_list (n-1)
a245508_list = f a000040_list a001105_list where
   f ps'@ (p:ps) xs'@(x:xs) = if p <= x then x : f ps xs' else f ps' xs

Module[{nn=60,ds},ds=2 Range[0,Ceiling[Sqrt[Prime[nn]]]]^2;Join[ {2},Table[ SelectFirst[ds,#>Prime[n]&],{n,2,nn}]]] (* Harvey P. Dale, Jan 07 2020 *)

a(n) = my(k=prime(n)+(n!=1)); while (! issquare(k/2), k+=2); k; \\ Michel Marcus, Jan 24 2022

A078327 Numbers k such that A078142(k) = A006530(k).

Original entry on oeis.org

2, 4, 6, 8, 12, 15, 16, 18, 24, 32, 36, 45, 48, 54, 64, 72, 75, 96, 105, 108, 110, 128, 135, 144, 162, 192, 216, 220, 225, 256, 288, 315, 324, 375, 384, 385, 405, 432, 440, 486, 512, 525, 550, 576, 648, 675, 735, 768, 864, 880, 935, 945, 972, 1024, 1100, 1125

Offset: 1

Jason Earls, Nov 24 2002

nonn

Numbers k such that the sum of the differences of the distinct prime factors p of k and the next square larger than p is equal to the largest prime factor of k.

Are there any other consecutive terms in this sequence other than 15,16 and 384,385?

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A006530, A078142, A078328, A145445.

is(k) = {if(k<2, return(0)); my(f=factor(k)[, 1]); sum(i=1, #f, (sqrtint(f[i])+1)^2-f[i]) == vecmax(f); } \\ Jinyuan Wang, Apr 17 2020

Offset changed to 1 by Jinyuan Wang, Apr 17 2020

A145447 a(n) = the smallest square (and/or cube) > n-th prime.

Original entry on oeis.org

4, 4, 8, 8, 16, 16, 25, 25, 25, 36, 36, 49, 49, 49, 49, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 121, 121, 121, 121, 121, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 196, 216, 216, 216, 225, 256, 256, 256, 256, 256, 256, 289, 289, 289, 289

Offset: 1

Zak Seidov, Oct 10 2008

nonn

a(n) = min(A145445(n), A145446(n) )= min ([A104103(n)]^2, [A104147(n)]^3)

A104103, A104147, A145445, A145446

Table[Min[{Ceiling[Prime[n]^(1/3)]^3,Ceiling[Prime[n]^(1/2)]^2}],{n,100}]

A229067 Sum of n-th prime and next perfect square.

Original entry on oeis.org

6, 7, 14, 16, 27, 29, 42, 44, 48, 65, 67, 86, 90, 92, 96, 117, 123, 125, 148, 152, 154, 160, 183, 189, 197, 222, 224, 228, 230, 234, 271, 275, 281, 283, 318, 320, 326, 332, 336, 369, 375, 377, 387, 389, 422, 424, 436, 448, 483, 485, 489, 495, 497, 507, 546, 552

Offset: 1

Vincenzo Librandi, Sep 25 2013

Primes in the sequence: 7, 29, 67, 197, 271, 281, 283, 389, 617, 631, 641, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A145445, A106044.

(Floor[Sqrt[#]] + 1)^2 + # &/@Prime[Range[80]]

a(n) = A000040(n) + A145445(n).

A229497 Product between n-th prime and next perfect square.

Original entry on oeis.org

8, 12, 45, 63, 176, 208, 425, 475, 575, 1044, 1116, 1813, 2009, 2107, 2303, 3392, 3776, 3904, 5427, 5751, 5913, 6399, 8300, 8900, 9700, 12221, 12463, 12947, 13189, 13673, 18288, 18864, 19728, 20016, 25181, 25519, 26533, 27547, 28223, 33908, 35084, 35476, 37436

Offset: 1

Vincenzo Librandi, Sep 25 2013

			63 is in the sequence because 7*9=63.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A145445, A106044.

(Floor[Sqrt[#]] + 1)^2 # &/@Prime[Range[80]]

a(n) = A000040(n) * A145445(n).

cp's OEIS Frontend

A145449 Numbers n such that A145445(n) = A145446(n).

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A014085 Number of primes between n^2 and (n+1)^2.

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A245508 Smallest double square (cf. A001105) greater than n-th prime.

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A078327 Numbers k such that A078142(k) = A006530(k).

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A145447 a(n) = the smallest square (and/or cube) > n-th prime.

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A229067 Sum of n-th prime and next perfect square.

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A229497 Product between n-th prime and next perfect square.

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