A242603 Largest divisor of n not divisible by 7. Remove factors 7 from n.

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 3, 22, 23, 24, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 5, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 1, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 10, 71, 72, 73, 74, 75, 76, 11

Offset

1,2

Comments

This is member p = 7 in the p-family of sequences (p a prime).

See A000265, A038502 and A132739 for primes 2, 3 and 5, also for formulas, programs and references.

As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

Links

Indranil Ghosh, Table of n, a(n) for n = 1..20000

Peter Bala, A note on the sequence of numerators of a rational function, 2019.

Formula

Multiplicative with a(p^e) = 1 if p = 7, else p^e.

Dirichlet g.f.: zeta(s-1)*7*(7^(s-1) - 1)/(7^s - 1).

a(n) = n/A268354(n).

From Peter Bala, Feb 21 2019: (Start)

a(n) = n/gcd(n,7^n).

O.g.f.: F(x) - 6*F(x^7) - 6*F(x^49) - 6*F(x^243) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,

Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) - (7^m - 1)( F(m,x^7) + F(m,x^49) + F(m,x^243) + ...), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.

Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence a(n) produces generating functions for the sequences (n^m*a(n))n>=1, m in Z. Some examples are given below. (End)

Sum_{k=1..n} a(k) ~ (7/16) * n^2. - Amiram Eldar, Nov 28 2022

Example

From Indranil Ghosh, Jan 31 2017: (Start)
For n = 12, the divisors of 12 are 1,2,3,4,6 and 12. The largest divisor not divisible by 7 is 12. So, a(12) = 12.
For n = 14, the divisors of 14 are 1,2,7 and 14. The largest divisor not divisible by 7 is 2. So, a(14) = 2. (End)
From Peter Bala, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (6*7)*G(x^7) - (6*49)*G(x^49) - (6*343)*G(x^343) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (6/7)*H(x^7) - (6/49)*H(x^49) - (6/343)*H(x^343) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (6/7^2)*L(x^7) - (6/49^2)*L(x^49) - (6/343^2)*L(x^343) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} (1/a(n))*x^n = L(x) + (6/7)*L(x^7) + (6/7)*L(x^49) + (6/7)*L(x^343) ... . (End)

Mathematica

Table[n/7^IntegerExponent[n, 7], {n, 80}] (* Alonso del Arte, Jun 18 2014 *)

Prog

(PARI) a(n) = f = factor(n);  for (i=1, #f~, if (f[i, 1]==7, f[i, 1]=1)); factorback(f); \\ Michel Marcus, Jun 18 2014
(PARI) a(n) = n \ 7^valuation(n, 7) \\ David A. Corneth, Feb 21 2019
(Python)
def A242603(n):
....for i in range(n, 0, -1):
........if n%i==0 and i%7!=0:
............return i # Indranil Ghosh, Jan 31 2017

Crossrefs

Cf. A000265, A038502, A132739, A214411.

Sequence in context: A269165 A319655 A328018 * A106608 A255692 A030106

Adjacent sequences: A242600 A242601 A242602 * A242604 A242605 A242606

Keyword

nonn,mult,easy

Author

Wolfdieter Lang, Jun 18 2014

Status

approved