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%I #63 Oct 06 2023 10:59:14
%S 1,1,1,2,5,1,7,4,3,5,11,2,13,7,5,8,17,3,19,10,7,11,23,4,25,13,9,14,29,
%T 5,31,16,11,17,35,6,37,19,13,20,41,7,43,22,15,23,47,8,49,25,17,26,53,
%U 9,55,28,19,29,59,10,61,31,21,32,65,11,67,34,23,35,71,12,73,37,25,38,77
%N a(n) = n / (gcd(n,2) * gcd(n,3)).
%C a(n+2) is absolute value of numerator of determinant of n X n matrix with M(i,j) = 2/(i(i+1)) if i=j otherwise 1. - _Alexander Adamchuk_, May 19 2006
%C Numerator of n/(n+6). - _Gerry Martens_, Aug 06 2015
%C In addition to being multiplicative, this sequence is also a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - _Peter Bala_, Feb 20 2019
%C Connected to the Diophantine equation x^2 + (x + 1)^2 + … + (x + k)^2 = C, C an integer, x >= 0, k >= 0. Rewritten it is (k + 1)*x^2 + k*(k+1)*x + k*(k + 1)*(2*k + 1)/6 = C. The existence of its solutions depends on gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6). - _Ctibor O. Zizka_, Oct 04 2023
%H Harry J. Smith, <a href="/A060789/b060789.txt">Table of n, a(n) for n=1..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2,0,0,0,0,0,-1).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Quasi-polynomial">Quasi-polynomial
%F G.f.: x*(1 + x + x^2 + 2*x^3 + 5*x^4 + x^5 + 5*x^6 + 2*x^7 + x^8 + x^9 + x^10)/(1 - x^6)^2.
%F Multiplicative with a(2^e)=2^(e-1), a(3^e)=3^(e-1), a(p^e)=p^e, p>3. - _Vladeta Jovovic_, Sep 09 2004
%F a(n) = Numerator[(-1)^(n+1)*Det[DiagonalMatrix[Table[2/(i(i+1))-1, {i,1,n-2}]]+1]], n>2. - _Alexander Adamchuk_, May 19 2006
%F a(n) divides n. a(6k) = k for integer k>0. a(p^k) = p^k for prime p>3 and integer k>0. - _Alexander Adamchuk_, Sep 20 2006
%F From _R. J. Mathar_, Apr 18 2011: (Start)
%F a(n) = A109047(n)/6.
%F Dirichlet g.f. zeta(s-1)*(1-1/2^s-2/3^s+2/6^s). (End)
%F a(n) = denominator((4*n-6)/n), n >= 2, with a(1) = 1. - _Johannes W. Meijer_, Dec 19 2012
%F a((2*n-1)*2^p) = A011782(p)*A146535(n), p >= 0. - _Johannes W. Meijer_, Feb 06 2013
%F a(n) = 2*a(n-6) - a(n-12) for n >= 12. - _Robert Israel_, Aug 06 2015
%F a(n) = gcd((n-1)*n*(n+1)/6, n). - _Lechoslaw Ratajczak_, Feb 16 2017
%F From _Peter Bala_, Feb 13 2019: (Start)
%F a(n) = n/gcd(n,n + 6) = n/gcd(n,6).
%F a(n) = n/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6.
%F a(n) is a quasi-polynomial in n: a(6*n+1) = 6*n + 1; a(6*n+2) = 3*n + 1; a(6*n+3) = 2*n + 1; a(6*n+4) = 3*n + 2; a(6*n+5) = 6*n + 5; a(6*n) = n.
%F a(n) = numerator(n/(n + 6)); a(n) = denominator((n + 6)/n).
%F (End)
%F Sum_{k=1..n} a(k) ~ 7*n^2/24. - _Vaclav Kotesovec_, Aug 09 2022
%F From _Ctibor O. Zizka_, Oct 04 2023: (Start)
%F For k >=0, a(k) = gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6).
%F If (k mod 6) = 0 or 4 then a(k) = (k + 1).
%F If (k mod 6) = 1 or 3 then a(k) = (k + 1)/2.
%F If (k mod 6) = 2 then a(k) = (k + 1)/3.
%F If (k mod 6) = 5 then a(k) = (k + 1)/6. (End)
%p a := proc(n): if n = 1 then 1 else denom((4*n-6)/n) fi: end: seq(a(n), n=1..77); # _Johannes W. Meijer_, Dec 19 2012
%t Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ 2/(i(i+1)) - 1, {i, 1, n-2} ] ] + 1 ], {n, 30} ]] (* _Alexander Adamchuk_, May 19 2006 *)
%t Table[Numerator[(n+3)/(n+2)/(n+1)/n],{n,60}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 17 2009 *)
%t Table[n/(GCD[n, 2] GCD[n, 3]), {n, 100}] (* _Wesley Ivan Hurt_, Aug 06 2015 *)
%t LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1}, {1,1,1,2,5,1,7,4,3,5,11,2}, 80] (* _Vincenzo Librandi_, Aug 07 2015 *)
%o (Sage) [lcm(n,6)/6 for n in range(1, 78)] # _Zerinvary Lajos_, Jun 07 2009
%o (PARI) { for (n=1, 1000, write("b060789.txt", n, " ", n / (gcd(n, 2) * gcd(n, 3))); ) } \\ _Harry J. Smith_, Jul 11 2009
%o (Magma) [n/(Gcd(n,2)*Gcd(n,3)) : n in [1..100]]; // _Wesley Ivan Hurt_, Aug 06 2015
%o (Magma) I:=[1,1,1,2,5,1,7,4,3,5,11,2]; [n le 12 select I[n] else 2*Self(n-6)-Self(n-12): n in [1..80]]; // _Vincenzo Librandi_, Aug 07 2015
%o (GAP) List([1..80],n->n/(Gcd(n,2)*Gcd(n,3))); # _Muniru A Asiru_, Feb 20 2019
%Y Cf. A011782, A109047, A146535, A220466.
%Y Cf. other sequences given by the formula n/gcd(n,k) = numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).
%K nonn,easy,mult
%O 1,4
%A _Len Smiley_, Apr 26 2001
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