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%I #41 Mar 13 2024 08:35:23
%S 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,
%T 27,4,29,30,31,32,33,34,5,36,37,38,39,40,41,6,43,44,45,46,47,48,1,50,
%U 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
%N Largest divisor of n not divisible by 7. Remove factors 7 from n.
%C This is member p = 7 in the p-family of sequences (p a prime).
%C See A000265, A038502 and A132739 for primes 2, 3 and 5, also for formulas, programs and references.
%C 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
%H Indranil Ghosh, <a href="/A242603/b242603.txt">Table of n, a(n) for n = 1..20000</a>
%H Peter Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function</a>, 2019.
%F Multiplicative with a(p^e) = 1 if p = 7, else p^e.
%F Dirichlet g.f.: zeta(s-1)*7*(7^(s-1) - 1)/(7^s - 1).
%F a(n) = n/A268354(n).
%F From _Peter Bala_, Feb 21 2019: (Start)
%F a(n) = n/gcd(n,7^n).
%F 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,
%F 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.
%F 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)
%F Sum_{k=1..n} a(k) ~ (7/16) * n^2. - _Amiram Eldar_, Nov 28 2022
%e From _Indranil Ghosh_, Jan 31 2017: (Start)
%e 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.
%e 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)
%e From _Peter Bala_, Feb 21 2019: (Start)
%e 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.
%e 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).
%e 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)).
%e 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)
%t Table[n/7^IntegerExponent[n, 7], {n, 80}] (* _Alonso del Arte_, Jun 18 2014 *)
%o (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
%o (PARI) a(n) = n \ 7^valuation(n, 7) \\ _David A. Corneth_, Feb 21 2019
%o (Python)
%o def A242603(n):
%o ....for i in range(n,0,-1):
%o ........if n%i==0 and i%7!=0:
%o ............return i # _Indranil Ghosh_, Jan 31 2017
%Y Cf. A000265, A038502, A132739, A214411.
%K nonn,mult,easy
%O 1,2
%A _Wolfdieter Lang_, Jun 18 2014
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