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

%I A309526 #9 Aug 14 2019 14:11:48
%S A309526 1,4,15,7,209,13,2911,97,901,181,564719,193,7865521,2521,6989,18817,
%T A309526 1525870529,2701,21252634831,37441,6779137,489061,4122901604639,37633,
%U A309526 274758906449,6811741,6575588101,1037623,11140078609864049,40321,155161278879431551
%N A309526 a(n) is the greatest divisor of A001353(n) that is coprime to A001353(m) for all positive integers m < n.
%C A309526 Analog of A178763 and A308949.
%C A309526 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.
%F A309526 a(n) = A306825(n) / gcd(A306825(n), n) if n != 2, 3.
%e A309526 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.
%o A309526 (PARI) T(n) = ([4, -1; 1, 0]^n)[2, 1]
%o A309526 b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
%o A309526 a(n) = if(isprime(n)&&!(12%n), b(n), b(n)/gcd(n, b(n)))
%Y A309526 Cf. A001353, A306825, A309040.
%Y A309526 Cf. A178763, A308949, A309525.
%K A309526 nonn
%O A309526 1,2
%A A309526 _Jianing Song_, Aug 06 2019