A077591 -id:A077591 - cp's OEIS frontend

A306257 a(n) = t for the minimal integer k > t such that k^2 mod n = t^2 is a perfect square.

0, 0, 1, 0, 2, 2, 2, 1, 0, 3, 3, 2, 3, 3, 1, 0, 4, 0, 4, 4, 2, 4, 4, 1, 0, 5, 3, 5, 5, 2, 5, 2, 4, 5, 1, 0, 6, 6, 5, 3, 6, 4, 6, 6, 2, 6, 6, 1, 0, 0, 7, 7, 7, 6, 3, 5, 5, 7, 7, 2, 7, 7, 1, 0, 4, 8, 8, 8, 7, 2, 8, 3, 8, 8, 5, 8, 2, 7, 8, 1, 0, 9, 9, 4, 6, 9, 7, 9, 9, 4, 3, 9, 8, 9, 7, 2, 9, 0, 1, 0, 10, 8, 10, 9, 4

Offset: 1

Alois P. Heinz, Feb 13 2019

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000290, A077591, A306271 (values of k).

a:= proc(n) local k; for k from (s-> `if`(s^2

a(n) = 0 <=> n > 0 and n in { A000290 } union { A077591 }.

A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 2, 20, 1, 6, 2, 1, 1, 1, 8, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Jan 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000290, A005117, A008833, A077591, A365331, A365332, A380163.

Cf. A002117, A013661, A078434.

f[p_, e_] := If[e == 1, 1, (p-1)*p^(2*Floor[e/2]-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1) * f[i, 1]^(2*(f[i, 2]\2)-1)));}

a(n) = A000010(A008833(n)).
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(2*floor(e/2)-1) if e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) / (zeta(2*s-1) * zeta(2*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3/2)/(zeta(2)*zeta(3)) = 1.32118019580177760682... .

A380164 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 1, 20, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 42, 6, 40

Offset: 1

Amiram Eldar, Jan 14 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000290, A002117, A077591, A268335, A350388, A351568, A365401, A380165.

f[p_, e_] := If[OddQ[e], 1, (p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}

a(n) = A000010(A350388(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is even, and 1 otherwise.
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s-1) - 1/p^(2*s) - 1/p^(3*s-2) + 1/p^(4*s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2) - 1/p^3 + 1/p^5) = 1.16404670858123447768... .

A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 4, 1, 4, 10, 2, 12, 6, 8, 1, 16, 1, 18, 4, 12, 10, 22, 8, 1, 12, 18, 6, 28, 8, 30, 16, 20, 16, 24, 1, 36, 18, 24, 16, 40, 12, 42, 10, 4, 22, 46, 2, 1, 1, 32, 12, 52, 18, 40, 24, 36, 28, 58, 8, 60, 30, 6, 1, 48, 20, 66, 16, 44, 24, 70, 4, 72

Offset: 1

Amiram Eldar, Jan 14 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000290, A013662, A077591, A268335, A350389, A351569, A365402, A374456, A380164.

f[p_, e_] := If[OddQ[e], (p-1)*p^(e-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1), 1));}

a(n) = A000010(A350389(n)).
a(n) >= 1, with equality if and only if n is either a square (A000290) or twice and odd square (A077591 \ {1}).
a(n) <= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.50115112192510092436... .

A137933 Least common multiple of n^2 and 2.

Original entry on oeis.org

2, 4, 18, 16, 50, 36, 98, 64, 162, 100, 242, 144, 338, 196, 450, 256, 578, 324, 722, 400, 882, 484, 1058, 576, 1250, 676, 1458, 784, 1682, 900, 1922, 1024, 2178, 1156, 2450, 1296, 2738, 1444, 3042, 1600, 3362, 1764, 3698, 1936, 4050, 2116, 4418, 2304, 4802, 2500, 5202, 2704

Offset: 1

William A. Tedeschi, Feb 29 2008

nonn

Cf. A016742, A077591, A109043, A245058.

LCM[Range[60]^2,2] (* Harvey P. Dale, Jun 06 2018 *)

a(n) = lcm(n^2, 2); \\ Michel Marcus, Mar 13 2018

a(n) = lcm(n^2, 2).
From R. J. Mathar, Mar 06 2008: (Start)
O.g.f.: -2x(1 + 6x^2 + x^4 + 2x^3 + 2x)/((-1+x)^3 * (x+1)^3).
a(2n) = A016742(n).
a(2n+1) = A077591(n). (End)
a(n) = n*A109043(n). - Michel Marcus, Mar 13 2018
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*Pi^2/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 = -A245058. (End)

A171642 Non-deficient numbers with odd sigma such that the sum of the even divisors is twice the sum of the odd divisors.

Original entry on oeis.org

18, 162, 450, 882, 1458, 2178, 2450, 3042, 4050, 5202, 6050, 6498, 7938, 8450, 9522, 11250, 13122, 15138, 17298, 19602, 22050, 24642, 27378, 30258, 33282, 36450, 39762, 43218, 46818, 50562, 54450, 58482, 61250, 62658, 66978, 71442, 76050, 80802, 85698

Offset: 1

Peter Luschny, Dec 14 2009

nonn

Numbers which are non-deficient (2n <= sigma(n)) [A023196] such that sigma(n) [A000203] is odd and the sum of the even divisors [A074400] is twice the sum of the odd divisors [A000593].

The sequence of terms which are not of the form 72*k^2 + 72*k + 18 starts: 2450, 6050, 8450, 61250, 120050, 151250, 211250, 296450.

			Divisors(18) = {1, 2, 3, 6, 9, 18}, sigma(18) = 39, and 2 + 6 + 18 = 2*(1 + 3 + 9).

Donovan Johnson, Table of n, a(n) for n = 1..1000
Peter Luschny, Zumkeller Numbers.

Cf. A000203, A023196, A074400, A000593.

Cf. A171641, A083207, A023196, A077591, A137933.

with(numtheory): A171642 := proc(n) local k,s,a;
s := sigma(n); a := add(k,k=select(x->type(x,odd),divisors(n)));
if 3*a = s and 2*n <= s and type(s,odd) then n else NULL fi end:

from sympy import divisors
A171642 = []
for n in range(1, 10**5):
    d = divisors(n)
    s = sum(d)
    if s % 2 and 2*n <= s and s == 3*sum([x for x in d if x % 2]):
        A171642.append(n)
# Chai Wah Wu, Aug 20 2014

A371124 a(n) is the least nonnegative integer y such that y^2 = x^2 - k*n for k and x where n > k >= 1 and n > x >= floor(sqrt(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 1, 0, 4, 5, 2, 6, 6, 1, 0, 8, 0, 9, 4, 2, 10, 11, 1, 0, 12, 3, 6, 14, 2, 15, 2, 4, 16, 1, 0, 18, 18, 5, 3, 20, 4, 21, 10, 2, 22, 23, 1, 0, 0, 7, 12, 26, 6, 3, 5, 8, 28, 29, 2, 30, 30, 1, 0, 4, 8, 33, 16, 10, 2, 35, 3, 36, 36, 5, 18, 2, 10

Offset: 1

Darío Clavijo, Mar 11 2024

nonn

a(A000290(n)) = 0.
a(A077591(n)) = 0.
a(A005563(n)) = 1.
For each n: k = A138191(n) and x = A306284(n).

			 n  | k | x | y^2 = x^2 - k*n  | y
------------------------------------
 1  | 1 | 1 | 0^2 = 1^2 - 1*1  | 0
 2  | 2 | 2 | 0^2 = 2^2 - 2*1  | 0
 11 | 1 | 6 | 5^2 = 6^2 - 1*11 | 5

Cf. A000040, A000290, A005563, A077591, A102781, A138191, A306284.

from sympy.core.power import isqrt
from sympy.ntheory.primetest import is_square
def a(n):
  x = isqrt(n)
  while True:
    for y2 in range(x**2-n, -1, -n):
      if is_square(y2): return isqrt(y2)
    x+=1
print([a(n) for n in range(1, 79)])

from itertools import count
def A371124(n):
    y, a = 0, {}
    for x in count(0):
        if y in a: return a[y]
        a[y] = x
        y = (y+(x<<1)+1)%n # Chai Wah Wu, Apr 25 2024

a(n) = floor(sqrt(A306284(n)^2 - n*A138191(n))).

a(A000040(n)) = A102781(n).

cp's OEIS Frontend

A306257 a(n) = t for the minimal integer k > t such that k^2 mod n = t^2 is a perfect square.

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A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

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A380164 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is a square.

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A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

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PARI

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A137933 Least common multiple of n^2 and 2.

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A171642 Non-deficient numbers with odd sigma such that the sum of the even divisors is twice the sum of the odd divisors.

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A371124 a(n) is the least nonnegative integer y such that y^2 = x^2 - k*n for k and x where n > k >= 1 and n > x >= floor(sqrt(n)).

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