A053187 -id:A053187 - cp's OEIS frontend

A245552 G.f.: Sum_{n>=0} (2n+1)x^(n^2+n+1).

0, 1, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19

Offset: 0

N. J. A. Sloane, Aug 02 2014

nonn

Related to g.f. for A053187.

Apart from signs and a factor of 2, this is the classical Jacobi theta-function theta'_1(q), see A002483.

Antti Karttunen, Table of n, a(n) for n = 0..10101

Cf. A002483, A053187, A198954.

Join[{0},Flatten[Table[Join[{n,PadRight[{},n,0]}],{n,1,19,2}]]] (* Harvey P. Dale, Dec 14 2014 *)

A198954(n) = { my(m); if(issquare(8*n + 1, &m), m, 0) }; \\ This function from Michael Somos
A245552(n) = if(!(n%2),0,A198954((n-1)/2)); \\ After Robert Israel's formula - Antti Karttunen, Jul 24 2017

a(2*n+1) = A198954(n), a(2*n) = 0.- Robert Israel, Aug 05 2014

A353295 Square nearest to the sum of the first n positive squares.

Original entry on oeis.org

0, 1, 4, 16, 25, 49, 100, 144, 196, 289, 400, 484, 625, 841, 1024, 1225, 1521, 1764, 2116, 2500, 2916, 3364, 3844, 4356, 4900, 5476, 6241, 6889, 7744, 8464, 9409, 10404, 11449, 12544, 13689, 14884, 16129, 17689, 19044, 20449, 22201, 23716, 25600, 27556, 29241

Offset: 0

Paolo Xausa, Jun 04 2022

			a(4) = 25 because the sum of the first 4 positive squares is 1 + 4 + 9 + 16 = 30, and the nearest square is 25.

Paolo Xausa, Table of n, a(n) for n = 0..9999

Cf. A000290, A000330, A053187, A351830, A354330.

nterms=100;Array[Round[Sqrt[#(#+1)(2#+1)/6]]^2&,nterms,0]

from math import isqrt
def a(n):
    s = n*(n+1)*(2*n+1)//6
    r = isqrt(s)
    d1, d2 = s-r**2, (r+1)**2-s
    return r**2 if d1 <= d2 else (r+1)**2
print([a(n) for n in range(45)]) # Michael S. Branicky, Jun 05 2022

a(n) = A053187(A000330(n)).

A351319 a(n) = floor(n/k), where k is the nearest square to n.

Original entry on oeis.org

1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 1

Joelle H. Kassir, Mar 18 2022

For all n != 2, a(n) is 0 when less than the nearest square, A053187(n), and is 1 otherwise.
From Jon E. Schoenfield, Mar 22 2022: (Start)
After the first two terms, the sequence consists of runs of 0's and 1's, with run lengths 1,3,2,4,3,5,4,6,5,7,6,8,... = A028242.
For m >= 1, there are 2m integers k whose nearest square is m^2, namely, the m-1 integers (in the interval [m^2-m+1, m^2-1]) for which k < m^2 (hence a(k) = 0), followed by the m+1 integers (in the interval [m^2, m^2+m]) for which k >= m^2 (hence a(k) = 1). (End)

			a(5) = floor(5/4) = 1.

Cf. A000194, A053187 (nearest square), A028242 (run lengths).

Cf. A267708 (essentially the same).

Table[Floor[n/Round[Sqrt[n]]^2], {n, 100}] (* Wesley Ivan Hurt, Mar 18 2022 *)

a(n) = if(n==2,2, my(r,s=sqrtint(n,&r)); r<=s); \\ Kevin Ryde, Mar 23 2022

import math
def a(n):
    k = math.isqrt(n)
    if n - k**2 > k: k += 1
    return n // k**2;
for n in range(1, 101):
    print("{}, ".format(a(n)), end="")

from math import isqrt
def A351319(n): return n if n <= 2 else int((k:=isqrt(n))**2+k-n+1 > 0) # Chai Wah Wu, Mar 26 2022

a(n) = floor(n/k), where k = round(sqrt(n))^2 = A053187(n).

a(n) = A267708(n) for n != 2.

A249077 Primes of the form n^2 + k such that n^2 - k is also prime, where -n < k < n.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 31, 41, 61, 67, 73, 79, 83, 89, 97, 103, 137, 139, 149, 151, 157, 181, 193, 199, 211, 223, 227, 239, 241, 271, 311, 317, 331, 337, 349, 373, 421, 433, 439, 443, 449, 461, 607, 619, 631, 643, 661, 691, 719, 739, 757, 811, 823, 829, 853, 859

Offset: 1

Arkadiusz Wesolowski, Oct 20 2014

nonn

Members of a pair (a, b) of primes such that a < b and the distances from a and b to the nearest square above a (or below b) are equal.
The only prime of the form n^2 + 1 (A002496) in the sequence is 5.
Is this sequence infinite?

			2^2-1=3, 2^2+1=5, both prime.
8^2-3=61, 8^2+3=67, both prime.

Cf. A047972, A053187.

lst:=[]; for m in [1..28] do r:=m*(m+1)+1; s:=(m+1)^2; for a in [r..s-1] do if IsPrime(a) then b:=2*s-a; if IsPrime(b) then Append(~lst, a); Append(~lst, b); end if; end if; end for; end for; Sort(lst);

g:= proc(t,m) if isprime(m+t) and isprime(m-t) then (m+t,m-t) else NULL fi end proc:
`union`(seq(map(g,{$1..n-1},n^2),n=2..100));
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list));
# Robert Israel, Oct 31 2014

for(n=1, 859, if(issquare(n), x=ps=n; until(issquare(x), x++); ns=x); if(isprime(n), if(n-ps

A prime p is in the sequence if and only if 2*A053187(p)-p is prime.

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A245552 G.f.: Sum_{n>=0} (2*n+1)*x^(n^2+n+1).

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A353295 Square nearest to the sum of the first n positive squares.

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A351319 a(n) = floor(n/k), where k is the nearest square to n.

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A249077 Primes of the form n^2 + k such that n^2 - k is also prime, where -n < k < n.

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A245552 G.f.: Sum_{n>=0} (2n+1)x^(n^2+n+1).