A247485 -id:A247485 - cp's OEIS frontend

A117767 a(n) is the difference between the smallest square greater than prime(n) and the largest square less than prime(n), where prime(n) = A000040(n) is the n-th prime number.

3, 3, 5, 5, 7, 7, 9, 9, 9, 11, 11, 13, 13, 13, 13, 15, 15, 15, 17, 17, 17, 17, 19, 19, 19, 21, 21, 21, 21, 21, 23, 23, 23, 23, 25, 25, 25, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 33, 33, 33, 35, 35, 35, 35, 35, 37, 37, 37, 37, 37, 37

Offset: 1

Odimar Fabeny, Apr 15 2006

From Reinhard Zumkeller, Sep 20 2014: (Start)
a(n) <= floor(2*sqrt(prime(n))) + 1 = A247485(n).
a(A247514(n)) = A247485(A247514(n)).
a(A247515(n)) < A247485(A247515(n)). (End)

			The 7th prime number is 17, which is between the consecutive squares 16 and 25, so a(7) = 25 - 16 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre's Conjecture.

Cf. A000040, A000006.

Cf. A096494, A247485, A247514, A247515.

a117767 = (+ 1) . (* 2) . a000006  -- Reinhard Zumkeller, Sep 20 2014

Mathematica
```
a[n_]:=2Floor[Sqrt[Prime[n]]]+1
```

{ forprime(p=2,200, f = floor(sqrt(p)) ; print1(2*f+1,",") ; ) ; } \\ R. J. Mathar, Apr 21 2006

a(n) = 2*A000006(n) + 1.

a(n) = 2*floor(sqrt(prime(n))) + 1. - R. J. Mathar, Apr 21 2006

More terms from R. J. Mathar, Apr 21 2006

Edited by Dean Hickerson, Jun 03 2006

A247514 Numbers k such that 2floor(sqrt(prime(k))) = floor(2sqrt(prime(k))).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 16, 19, 20, 23, 24, 26, 27, 28, 29, 31, 32, 35, 36, 40, 41, 42, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 62, 67, 68, 73, 74, 75, 79, 80, 81, 86, 87, 88, 89, 93, 94, 95, 96, 100, 101, 106, 107, 108, 109, 115, 116, 117, 118, 123, 124

Offset: 1

Reinhard Zumkeller, Sep 20 2014

nonn

A117767(a(n)) = A247485(a(n)); complement of A247515.

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

Cf. A117767, A247485, A000196, A000040, A000006, A247515.

a247514 n = a247514_list !! (n-1)
a247514_list = filter (\x -> a117767 x == a247485 x) [1..]

A247514Q[k_]:=With[{r=Sqrt[Prime[k]]},2Floor[r]==Floor[2r]];
Select[Range[200],A247514Q] (* Paolo Xausa, Oct 23 2023 *)

isok(k) = my(p=prime(k)); 2*sqrtint(p) == sqrtint(4*p); \\ Michel Marcus, Apr 29 2023

A247515 Numbers k such that 2floor(sqrt(prime(k))) < floor(2sqrt(prime(k))).

Original entry on oeis.org

2, 4, 6, 9, 11, 14, 15, 17, 18, 21, 22, 25, 30, 33, 34, 37, 38, 39, 43, 44, 47, 48, 53, 54, 59, 60, 61, 63, 64, 65, 66, 69, 70, 71, 72, 76, 77, 78, 82, 83, 84, 85, 90, 91, 92, 97, 98, 99, 102, 103, 104, 105, 110, 111, 112, 113, 114, 119, 120, 121, 122, 127

Offset: 1

Reinhard Zumkeller, Sep 20 2014

nonn

A117767(a(n)) < A247485(a(n)); complement of A247514.

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

Cf. A117767, A247485, A000196, A000040, A000006, A247514.

a247515 n = a247515_list !! (n-1)
a247515_list = filter (\x -> a117767 x < a247485 x) [1..]

A247515Q[k_]:=With[{r=Sqrt[Prime[k]]},2Floor[r]A247515Q] (* Paolo Xausa, Oct 23 2023 *)

A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 17, 17, 19, 21, 23, 25, 25, 27, 27, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 41, 41, 41, 41, 43, 45, 45, 47, 47, 49, 49, 51, 51, 51, 53, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 61, 61, 63, 63, 65, 65, 65, 65, 67, 67, 67, 69, 71, 71, 71, 71, 73, 73, 75, 75, 75, 75, 77

Offset: 1

Robin Visser, Apr 25 2023

nonn

Two elliptic curves over a finite field F_q are isogenous if and only if they have the same trace of Frobenius, or equivalently, have the same number of points over F_q.

Thus, by the Hasse bound, a(n) is the number of integers with absolute value bounded by 2*sqrt(prime(n)).

			For n = 1, the a(1) = 5 isogeny classes of elliptic curves are parametrized by the 5 possible values for the trace of Frobenius: -2, -1, 0, 1, 2.
For n = 2, the a(2) = 7 isogeny classes of elliptic curves are parametrized by the 7 possible values for the trace of Frobenius: -3, -2, -1, 0, 1, 2, 3.

Robin Visser, Table of n, a(n) for n = 1..10000
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
J. H. Silverman, The Arithmetic of Elliptic Curves, Second edition. Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009.

Cf. A247485, A362198, A362201, A362243, A364681.

[2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];

2Floor[2Sqrt[Prime[Range[100]]]]+1 (* Paolo Xausa, Oct 23 2023 *)

PARI
```
a(n) = 2*sqrtint(4*prime(n)) + 1;
```

a(n) = 2*floor(2*sqrt(prime(n))) + 1.

a(n) = 2*A247485(n) - 1.

A347038 Primes p such that there are no solutions to d(k+p) = sigma(k).

Original entry on oeis.org

29, 37, 41, 53, 67, 89, 101, 109, 113, 127, 137, 151, 157, 173, 181, 197, 227, 229, 233, 257, 269, 277, 281, 293, 313, 349, 373, 389, 401, 409, 421, 439, 461, 557, 587, 593, 601, 613, 617, 641, 643, 653, 661, 673, 677, 701, 709, 739, 761, 773, 787, 821, 829

Offset: 1

Angad Singh, Aug 12 2021

nonn

If p is not in the sequence and d(k+p) = sigma(k), then k <= 1+2*sqrt(p). Proof: We have d(m) <= 2*sqrt(m) (see formula in A000005), so 2*sqrt(k+p) >= d(k+p) = sigma(k) >= k+1 (if k > 1). After squaring and simplifying, we get k <= 1+2*sqrt(p). - Pontus von Brömssen, Aug 20 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A247485.

filter:= proc(p) isprime(p) and not ormap(k -> numtheory:-tau(k+p) = numtheory:-sigma(k), [$1 .. 1 + 2*isqrt(p)]) end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Aug 06 2025

from sympy import divisor_count as d, divisor_sigma as sigma, primerange
from math import isqrt
def A347038_list(pmax):
    a = []
    for p in primerange(2, pmax + 1):
        if not any(d(k + p) == sigma(k) for k in range(1, 2 + isqrt(4 * p))):
            a.append(p)
    return a  # Pontus von Brömssen, Aug 20 2021

cp's OEIS Frontend

A117767 a(n) is the difference between the smallest square greater than prime(n) and the largest square less than prime(n), where prime(n) = A000040(n) is the n-th prime number.

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A247514 Numbers k such that 2*floor(sqrt(prime(k))) = floor(2*sqrt(prime(k))).

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A247515 Numbers k such that 2*floor(sqrt(prime(k))) < floor(2*sqrt(prime(k))).

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Haskell

Mathematica

A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A347038 Primes p such that there are no solutions to d(k+p) = sigma(k).

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Author

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Comments

Links

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Programs

Maple

Python

A247514 Numbers k such that 2floor(sqrt(prime(k))) = floor(2sqrt(prime(k))).

A247515 Numbers k such that 2floor(sqrt(prime(k))) < floor(2sqrt(prime(k))).