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

%I A309525 #13 Aug 02 2024 11:19:13
%S A309525 1,3,10,11,109,1,1189,119,1297,131,141481,59,1543321,1429,3089,14159,
%T A309525 183642229,433,2003229469,14041,1837837,170039,238367471761,7079,
%U A309525 23854956949,1854841,2186871697,1670761,309400794703549,12871,3375045015828949,200477279
%N A309525 a(n) is the greatest divisor of A006190(n) that is coprime to A006190(m) for all positive integers m < n.
%C A309525 Analog of A178763 and A308949.
%H A309525 Robert Israel, <a href="/A309525/b309525.txt">Table of n, a(n) for n = 1..1930</a>
%F A309525 a(n) = A253807(n) / gcd(A253807(n), n) if n != 6, 13.
%e A309525 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.
%p A309525 A6190:= proc(n) option remember; 3*procname(n-1)+procname(n-2) end proc:
%p A309525 A6190(0):= 0: A6190(1):= 1:
%p A309525 f:= proc(n) local k,i,g;
%p A309525   k:= A6190(n);
%p A309525   for i from 1 to n-1 do
%p A309525     g:= igcd(k,A6190(i));
%p A309525     while g > 1 do
%p A309525       k:= k/g;
%p A309525       g:= igcd(k,A6190(i));
%p A309525     od;
%p A309525   od;
%p A309525   k
%p A309525 end proc:
%p A309525 map(f, [$1..40]); # _Robert Israel_, Aug 02 2024
%o A309525 (PARI) T(n) = ([3, 1; 1, 0]^n)[2, 1]
%o A309525 b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
%o A309525 a(n) = if(isprime(n)&&!(13%n), 1543321, if(n!=6, b(n)/gcd(n, b(n)), 1))
%Y A309525 Cf. A006190, A253807.
%Y A309525 Cf. A178763, A308949, A309526.
%K A309525 nonn,look
%O A309525 1,2
%A A309525 _Jianing Song_, Aug 06 2019