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

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, 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, 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

Offset: 1

Author

Barry Wells (wells.barry(AT)gmail.com), Sep 25 2009

Keywords

• mult
• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• Barry Wells, Table of n, a(n) for n = 1..1024 [a(392) and a(704) corrected by Sean A. Irvine]

Crossrefs

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 }.

Largest divisor of n coprime to a prime factor of 30: A000265 (2), A038502 (3), A132739 (5).

Cf. A355582.

Programs

Mathematica

a[n_] := n / Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100] (* Amiram Eldar, Jul 10 2022 *)

PARI

a(n)=n>>valuation(n,2)/3^valuation(n,3)/5^valuation(n,5) \\ Charles R Greathouse IV, Jul 16 2017

Formula

From Amiram Eldar, Jul 10 2022: (Start)

Multiplicative with a(p^e) = p^e if p >= 7 and 1 otherwise.

a(n) = n/A355582(n). (End)

Sum_{k=1..n} a(k) ~ (5/24) * n^2. - Amiram Eldar, Nov 28 2022

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