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a(n) is the smallest x for which the following quotient is an integer: (sigma(x) + ... + sigma(x+n-1))/sigma(x+(x+1)+ ... + (x+n-1)), i.e., sum(sigma(j))/sigma(sum(j)) for n terms summed up was integer.
2

%I #14 Jul 29 2017 04:11:50

%S 1,1,424,7,13980,2,805675056,15662957033,7,42,48957

%N a(n) is the smallest x for which the following quotient is an integer: (sigma(x) + ... + sigma(x+n-1))/sigma(x+(x+1)+ ... + (x+n-1)), i.e., sum(sigma(j))/sigma(sum(j)) for n terms summed up was integer.

%C Some other values: a(15)=298, a(19)=97.

%C a(12) > 8*10^10. - _Giovanni Resta_, May 07 2017

%e n=6: sigma(2) + sigma(3) + ... + sigma(7) = 3 + 4 + 7 + 6 + 12 + 8 = 40 = sigma(2 + 3 + ... + 7) = sigma(27) = 1 + 3 + 9 + 27 = 40; quotient = 1.

%t g[x_, k_] := Apply[Plus, Table[sg[x+j], j, 0, k-1}]]/ sg[Apply[Plus, Table[x+j, j, 0, k-1}]]] Table[fla=1;Do[s=g[n, h]; If[IntegerQ[s]&&Equal[fla, 1], Print[{n, h}];fla=0], {n, 1, 10000000}], {h, 1, 6}]

%Y Cf. A000203, A091287-A091293.

%K hard,nonn

%O 1,3

%A _Labos Elemer_, Feb 17 2004

%E a(7)-a(11) from _Giovanni Resta_, May 07 2017