OFFSET
1,24
REFERENCES
Senechal, Marjorie. "Introduction to lattice geometry." In M. Waldschmidt et al., eds., From Number Theory to Physics, pp. 476-495. Springer, Berlin, Heidelberg, 1992. See Cor. 3.7.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
FORMULA
a(n + 168) = a(n). - David A. Corneth, Nov 05 2018
Conjectures from Colin Barker, Nov 05 2018: (Start)
G.f.: x^6*(1 + 2*x + 4*x^2 + 5*x^3 + 7*x^4 + 8*x^5 + 10*x^6 + 9*x^7 + 9*x^8 + 6*x^9 + 6*x^10 + 3*x^11 + 3*x^12) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = -a(n-1) - 2*a(n-2) - 2*a(n-3) - 3*a(n-4) - 3*a(n-5) - 3*a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + 2*a(n-11) + 3*a(n-12) + 3*a(n-13) + 3*a(n-14) + 2*a(n-15) + 2*a(n-16) + a(n-17) + a(n-18) for n>18.
(End)
From David A. Corneth, Nov 05 2018: (Start)
The above conjectures are true. The sequence is periodic with period 168. Let f(x) be the g.f. above. Then f(x + 168) = f(x).
The expression for a(n) holds for 19 <= n <= 1000, more than sufficient for a proof. (End)
EXAMPLE
a(24) = 2 because 24 is divisible 6 and 8, i.e., by 2 of the numbers in {6, 7, 8}. David A. Corneth, Nov 05 2018
MAPLE
d3:=proc(n) local c; c:=0;
if (n mod 6)=0 then c:=c+1; fi;
if (n mod 7)=0 then c:=c+1; fi;
if (n mod 8)=0 then c:=c+1; fi; c; end;
[seq(d3(n), n=1..120)];
MATHEMATICA
a[n_]:=Sum[If[Mod[n, i]==0, 1, 0], {i , 6, 8}]; Array[a, 120] (* Stefano Spezia, Nov 05 2018 *)
Table[Total[Boole[Divisible[n, {6, 7, 8}]]], {n, 120}] (* Harvey P. Dale, Nov 09 2022 *)
PROG
(PARI) a(n) = sum(i = 6, 8, !(n%i)) \\ David A. Corneth, Nov 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 04 2018
STATUS
approved