A309526 - OEIS

A309526

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

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..31.

FORMULA

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

EXAMPLE

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.

PROG

(PARI) 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)))

CROSSREFS

Cf. A001353, A306825, A309040.

Cf. A178763, A308949, A309525.

Sequence in context: A328839 A143899 A154068 * A126601 A284119 A290377

Adjacent sequences: A309523 A309524 A309525 * A309527 A309528 A309529

KEYWORD

nonn

AUTHOR

Jianing Song, Aug 06 2019

STATUS

approved