A308949 -id:A308949 - 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.

A309526 a(n) is the greatest divisor of A001353(n) that is coprime to A001353(m) for all positive integers m < n.

Original entry on oeis.org

1, 4, 15, 7, 209, 13, 2911, 97, 901, 181, 564719, 193, 7865521, 2521, 6989, 18817, 1525870529, 2701, 21252634831, 37441, 6779137, 489061, 4122901604639, 37633, 274758906449, 6811741, 6575588101, 1037623, 11140078609864049, 40321, 155161278879431551

Offset: 1

Jianing Song, Aug 06 2019

nonn

Analog of A178763 and A308949.

Let b(n) = A309040(n)*gcd(A309040(n),n), then for n > 3: a(n) = b(2n) for even n and b(n)*b(2n) for odd n. It seems highly impossible that b(n) = 1 holds for n > 3, so it seems that only even-indexed terms can be primes.

			A001353(6) = 780 = 2^2 * 3 * 5 * 13. We have 2 divides A001353(2) = 2 and 3, 5 divides A001353(3) = 15, but A001353(m) is coprime to 13 for all 1 <= m < 6, so a(6) = 13.

Cf. A001353, A306825, A309040.

Cf. A178763, A308949, A309525.

T(n) = ([4, -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)&&!(12%n), b(n), b(n)/gcd(n, b(n)))

a(n) = A306825(n) / gcd(A306825(n), n) if n != 2, 3.

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.

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