Sofia Lacerda - cp's OEIS frontend

A353701 Denominator of squared radius of smallest circle passing through exactly n integral points.

4, 18, 2, 18, 4, 242, 2, 18, 4, 242, 2, 98, 4, 18, 2, 578, 4, 578, 2, 242, 242, 98, 2, 18, 98, 18, 2, 722, 4, 98, 2, 162

Offset: 2

Sofia Lacerda, May 04 2022

Schinzel proved such a circle always exists, and the square of the radius of a circle passing through 3 integral points is always rational so the sequence is well-defined.

			For n=3 a minimal circle is (x - 1/6)^2 + (y - 1/6)^2 = 25/18.

S. S. Lacerda, schinzel.py
E. Pegg, Lattice Circles
Jim Randell, A collection of minimal radius lattice circles (github)
C. Schinzel, Sur l'existence d'un cercle passant par un nombre donné de points aux coordonnées entières, Enseignement Math, vol. 4, pp. 71-72, 1958.

Numerators are A353700.

Data corrected by Sean A. Irvine, Jul 19 2022

a(29)-a(33) from Jim Randell, Jan 10 2023

A353700 Numerator of squared radius of smallest circle passing through exactly n integral points.

Original entry on oeis.org

1, 25, 1, 625, 25, 138125, 5, 4225, 625, 801125, 25, 1221025, 15625, 105625, 65, 185870425, 4225, 185870425, 625, 29641625, 29641625, 30525625, 325, 17850625, 35409725, 1221025, 15625, 3159797225, 105625, 763140625, 1105, 1346691125

Offset: 2

Sofia Lacerda, May 04 2022

Schinzel proved such a circle always exists, and the square of the radius of a circle passing through 3 integral points is always rational so the sequence is well-defined.

			For n=3 a minimal circle is (x - 1/6)^2 + (y - 1/6)^2 = 25/18.

Sean A. Irvine, Java program (github)
S. S. Lacerda, schinzel.py
E. Pegg, Lattice Circles
Jim Randell, A collection of minimal radius lattice circles (github)
C. Schinzel, Sur l'existence d'un cercle passant par un nombre donné de points aux coordonnées entières, Enseignement Math, vol. 4, pp. 71-72, 1958.

Denominators are A353701.

Data corrected by Sean A. Irvine, Jul 17 2022

a(29)-a(33) from Jim Randell, Jan 10 2023

A346587 Value of k for incrementally largest value of k/A002322(k), where A002322 is the Carmichael function.

Original entry on oeis.org

1, 2, 6, 8, 12, 24, 60, 80, 120, 240, 504, 1040, 1092, 1170, 1260, 1365, 1456, 1560, 1638, 1680, 1820, 1872, 2184, 2340, 2520, 2730, 3120, 3276, 3640, 4095, 4368, 4680, 5040, 5460, 6552, 7280, 8190, 9360, 10920, 13104, 16380, 21840, 32760, 65520

Offset: 1

Sofia Lacerda, Jul 24 2021

Cf. A002322.

max=0;lst={};Do[t=k/CarmichaelLambda@k;If[t>max,AppendTo[lst,k];max=t],{k,100000}];lst (* Giorgos Kalogeropoulos, Jul 26 2021 *)

f(n) = lcm(znstar(n)[2]); \\ A002322
lista(nn) = {my(m=0, x, list=List()); for (n=1, nn, if ((x = n/f(n)) > m, m = x; listput(list, n));); Vec(list);} \\ Michel Marcus, Aug 04 2021

A344202 Primes p such that gcd(ord_p(2), ord_p(3)) = 1.

Original entry on oeis.org

683, 599479, 108390409, 149817457, 666591179, 2000634731, 4562284561, 14764460089, 24040333283, 2506025630791, 5988931115977

Offset: 1

Sofia Lacerda, May 11 2021

'ord_p' here means the multiplicative order, not to be confused with the p-adic order that is also often denoted by ord_p.

Related to Diophantine equations of the form (2^x-1)*(3^y-1) = n*z^2.

Sofia Lacerda, C++ program (github).
MathOverflow, Solve this Diophantine equation (2^x-1)(3^y-1)=2z^2
Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021.
Wikipedia, Multiplicative order.
Wikipedia, P-adic_valuation.

Cf. A014664, A062117.

Select[Range[10^6], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[3, #]] &] (* Amiram Eldar, May 11 2021 *)

isok(p) = isprime(p) && (gcd(znorder(Mod(2, p)), znorder(Mod(3, p))) == 1); \\ Michel Marcus, May 11 2021

from sympy.ntheory import n_order
from sympy import gcd, nextprime
A344202_list, p = [], 5
while p < 10**9:
    if gcd(n_order(2,p),n_order(3,p)) == 1:
        A344202_list.append(p)
    p = nextprime(p) # Chai Wah Wu, May 12 2021

a(3)-a(5) from Michel Marcus, May 11 2021
a(6)-a(8) from Amiram Eldar, May 11 2021
a(9) from Daniel Suteu, May 16 2021
a(10) from Sofia Lacerda, Jul 07 2021
a(11) from Sofia Lacerda, Aug 03 2021

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User: Sofia Lacerda

A353701 Denominator of squared radius of smallest circle passing through exactly n integral points.

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A353700 Numerator of squared radius of smallest circle passing through exactly n integral points.

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A346587 Value of k for incrementally largest value of k/A002322(k), where A002322 is the Carmichael function.

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A344202 Primes p such that gcd(ord_p(2), ord_p(3)) = 1.

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Mathematica

PARI

Python

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