A253807 -id:A253807 - cp's OEIS frontend

A309525 a(n) is the greatest divisor of A006190(n) that is coprime to A006190(m) for all positive integers m < n.

1, 3, 10, 11, 109, 1, 1189, 119, 1297, 131, 141481, 59, 1543321, 1429, 3089, 14159, 183642229, 433, 2003229469, 14041, 1837837, 170039, 238367471761, 7079, 23854956949, 1854841, 2186871697, 1670761, 309400794703549, 12871, 3375045015828949, 200477279

Offset: 1

Jianing Song, Aug 06 2019

Analog of A178763 and A308949.

			A006190(12) = 467280 = 2^4 * 3^2 * 5 * 11 * 59. We have 2, 3, 5 divides A006190(6) = 360 and 11 divides A006190(3) = 11, but A006190(m) is coprime to 59 for all 1 <= m < 12, so a(12) = 59.

Robert Israel, Table of n, a(n) for n = 1..1930

Cf. A006190, A253807.

Cf. A178763, A308949, A309526.

A6190:= proc(n) option remember; 3*procname(n-1)+procname(n-2) end proc:
A6190(0):= 0: A6190(1):= 1:
f:= proc(n) local k,i,g;
  k:= A6190(n);
  for i from 1 to n-1 do
    g:= igcd(k,A6190(i));
    while g > 1 do
      k:= k/g;
      g:= igcd(k,A6190(i));
    od;
  od;
  k
end proc:
map(f, [$1..40]); # Robert Israel, Aug 02 2024

T(n) = ([3, 1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(isprime(n)&&!(13%n), 1543321, if(n!=6, b(n)/gcd(n, b(n)), 1))

a(n) = A253807(n) / gcd(A253807(n), n) if n != 6, 13.

A309041 Irregular table read by rows: Let P(n,x) be the (monic) minimal polynomial of 2i*cos(Pi/n), where i = sqrt(-1) is the imaginary unit, then a(n,k) = [x^(2k)] P(n,x), n >= 3.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 6, 5, 1, 2, 4, 1, 1, 9, 6, 1, 5, 5, 1, 1, 15, 35, 28, 9, 1, 1, 4, 1, 1, 21, 70, 84, 45, 11, 1, 7, 14, 7, 1, 1, 24, 26, 9, 1, 2, 16, 20, 8, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 3, 9, 6, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1

Offset: 3

Jianing Song, Jul 08 2019

For n >= 3, it is easy to see that [x^(2k+1)] P(n,x) = 0, so they are omitted.
Row n (n >= 3) has length A023022(n) + 1 = phi(n)/2 + 1.
Let {U(n,x)} be defined as: U(0,x) = 0, U(1,x) = 1, U(n,x) = x*U(n-1,x) + U(n-2,x) for n >= 2, then U(n,x) = Product_{k|n, k>=2} P(k,x) for n > 0, because U(n,x) = Product_{m=1..n-1} (x - 2i*cos(Pi*m/n)) for n > 0.

			P(1,x) = x^2 + 4;
P(2,x) = x;
P(3,x) = x^2 + 1;
P(4,x) = x^2 + 2;
P(5,x) = x^4 + 3x^2 + 1;
P(6,x) = x^2 + 3;
P(7,x) = x^6 + 5x^4 + 6x^2 + 1;
P(8,x) = x^4 + 4x^2 + 2;
P(9,x) = x^6 + 6x^4 + 9x^2 + 1;
P(10,x) = x^4 + 5x^2 + 5;
...

Cf. A232624.
Cf. P(n,k): A061446 (k=1), A008555 (k=2), A253807 (k=3);
Cf. also A023022, A008683.

ro[n_] := (P = CoefficientList[p = MinimalPolynomial[2*I*Cos[Pi/n], x], x^2]; P); Flatten[Table[ro[n], {n, 3, 30}]]

U(n) = sum(i=0, (n-1)/2,binomial(n,2*i+1)*(poly/2)^(n-2*i-1)*((poly^2+4)/4)^i)
P(n) = if(n==1, poly^2+4, my(v=divisors(n)); prod(i=1, #v, U(n/v[i])^moebius(v[i])))
a(n,k) = polcoeff(P(n),2*k)

P(n,x) = Product_{0<=m<=n, gcd(m, n)=1} (x - 2i*cos(Pi*m/n)).
Equivalently, P(n,x) = Product_{0<=m<=n/2, gcd(m, n)=1} (x^2 + 4*cos(Pi*m/n)) for n != 2. This shows that all terms are positive.
P(n,x) = Product_{k|n} U(n/k,x)^mu(k), mu = A008683.
Let MPR2(n,x) be the (monic) minimal polynomial of 2*cos(2*Pi/n) as defined in A232624, then: for even n > 2, P(n,x) = MPR2(2n,i*x)*(-1)^A023022(n); for odd n, P(n,x) = MPR2(n,i*x)*MPR2(2n,i*x)*(-1)^A023022(n), i = sqrt(-1).
For n > 2, P(n,x) = MPR2(n,-x^2-2)*(-1)^A023022(n).
For n > 1, P(n,1) = A061446(n), P(n,2) = A008555(n), P(n,3) = A253807(n), ...
For even n > 2, a(n,k) = (-1)^(A023022(n)-k)*A232624(2n,2k).

cp's OEIS Frontend

A309525 a(n) is the greatest divisor of A006190(n) that is coprime to A006190(m) for all positive integers m < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula