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A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

%I A165725 #33 May 17 2024 19:47:31
%S A165725 1,1,1,1,1,1,7,1,1,1,11,1,13,7,1,1,17,1,19,1,7,11,23,1,1,13,1,7,29,1,
%T A165725 31,1,11,17,7,1,37,19,13,1,41,7,43,11,1,23,47,1,49,1,17,13,53,1,11,7,
%U A165725 19,29,59,1,61,31,7,1,13,11,67,17,23,7,71,1,73,37,1,19,77,13,79,1,1,41,83,7
%N A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.
%C A165725 This is the sequence of the largest divisor of n which is coprime to 30. The product of the first 3 prime numbers is 2*3*5=30. This sequence gives the largest factor of n which does not include 2, 3 or 5 in its prime factorization.
%H A165725 Barry Wells, <a href="/A165725/b165725.txt">Table of n, a(n) for n = 1..1024</a> [a(392) and a(704) corrected by Sean A. Irvine]
%F A165725 From _Amiram Eldar_, Jul 10 2022: (Start)
%F A165725 Multiplicative with a(p^e) = p^e if p >= 7 and 1 otherwise.
%F A165725 a(n) = n/A355582(n). (End)
%F A165725 Sum_{k=1..n} a(k) ~ (5/24) * n^2. - _Amiram Eldar_, Nov 28 2022
%F A165725 Dirichlet g.f.: zeta(s-1)*(2^s-2)*(3^s-3)*(5^s-5)/((2^s-1)*(3^s-1)*(5^s-1)). - _Amiram Eldar_, Jan 04 2023
%e A165725 The largest factor of 1, 2, 3, 4, 5 and 6 not including the primes 2, 3 and 5 is 1. 7 is prime and therefore its sequence value is 7. For p > 5, p prime, gives a(p) = p. As 14 = 2*7, a(14)= 7. As 98 = 2*7*7, a(98)= 49.
%t A165725 a[n_] := n / Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100] (* _Amiram Eldar_, Jul 10 2022 *)
%o A165725 (PARI) a(n)=n>>valuation(n,2)/3^valuation(n,3)/5^valuation(n,5) \\ _Charles R Greathouse IV_, Jul 16 2017
%Y A165725 A051037 gives the smooth five numbers, numbers whose prime divisor only include 2, 3 and 5. A132740 gives the largest divisor of n coprime to 10. A065330 gives a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.
%Y A165725 Largest divisor of n coprime to a prime factor of 30: A000265 (2), A038502 (3), A132739 (5).
%Y A165725 Cf. A355582.
%K A165725 mult,nonn,easy
%O A165725 1,7
%A A165725 Barry Wells (wells.barry(AT)gmail.com), Sep 25 2009