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A374840 a(n) is the greatest m > 0 such that the n-th row of Pascal's triangle (A007318) contains a multiple of k for k = 1..m.

%I A374840 #17 Sep 04 2024 12:19:18
%S A374840 1,1,2,1,4,2,6,1,2,4,10,3,12,6,4,1,16,2,18,4,6,10,22,3,4,12,2,6,28,15,
%T A374840 30,1,10,16,6,8,36,18,12,7,40,6,42,10,8,22,46,3,6,4,16,12,52,2,10,7,
%U A374840 18,28,58,15,60,30,8,1,12,10,66,16,22,24,70,8,72,36
%N A374840 a(n) is the greatest m > 0 such that the n-th row of Pascal's triangle (A007318) contains a multiple of k for k = 1..m.
%C A374840 The sequence A006093 appears to give the fixed points of this sequence.
%H A374840 Alois P. Heinz, <a href="/A374840/b374840.txt">Table of n, a(n) for n = 0..10000</a>
%e A374840 For n = 6: the sixth row of Pascal's triangle is 1, 6, 15, 20, 15, 6, 1; it contains a multiple of 1 (1), of 2 (6), of 3 (6), of 4 (20), of 5 (15), of 6 (6), but not of 7, so a(6) = 6.
%p A374840 A374840 := proc(n)
%p A374840     local dvsn ,m,a;
%p A374840     if n = 0 then
%p A374840         return 1;
%p A374840     end if;
%p A374840     dvsn := {} ;
%p A374840     for m from 1 to (n+2)/2 do
%p A374840         binomial(n,m) ;
%p A374840         dvsn := dvsn union numtheory[divisors](%) ;
%p A374840     end do:
%p A374840     for a from 1 do
%p A374840         if not a in dvsn then
%p A374840             return a-1 ;
%p A374840         end if;
%p A374840     end do:
%p A374840 end proc:
%p A374840 seq(A374840(n),n=0..40) ; # _R. J. Mathar_, Jul 30 2024
%p A374840 # second Maple program:
%p A374840 a:= proc(n) local k, s; s:= {seq(binomial(n,k), k=0..n/2)};
%p A374840       for k while ormap(x-> irem(x, k)=0, s) do od: k-1
%p A374840     end:
%p A374840 seq(a(n), n=0..73);  # _Alois P. Heinz_, Sep 04 2024
%t A374840 a[n_] := If[n == 0, 1, Module[{dd, m, k}, dd = {}; For[m = 1, m <= (n + 2)/2, m++, dd = Union[dd, Divisors[Binomial[n, m]]]]; For[k = 1, True, k++, If[FreeQ[dd, k], Return[k - 1]]]]];
%t A374840 Table[a[n], {n, 0, 73}] (* _Jean-François Alcover_, Sep 04 2024, after _R. J. Mathar_ *)
%o A374840 (PARI) a(n) = { my (b = binomial(n)[1..(n+2)\2]); for (m = 2, oo, ok = 0; for (i = 1, #b, if (b[i] % m==0, next(2); ); ); return (m-1); ); }
%Y A374840 Cf. A006093, A007318, A249151.
%K A374840 nonn
%O A374840 0,3
%A A374840 _Rémy Sigrist_, Jul 22 2024