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A185283 Least k such that sigma(1) + sigma(2) + sigma(3) +...+ sigma(k) >= n.

%I A185283 #25 Dec 27 2024 08:45:32
%S A185283 0,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,
%T A185283 7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,
%U A185283 9,9,10,10,10,10,10,10,10,10,10,10,10,10,10
%N A185283 Least k such that sigma(1) + sigma(2) + sigma(3) +...+ sigma(k) >= n.
%H A185283 Alois P. Heinz, <a href="/A185283/b185283.txt">Table of n, a(n) for n = 0..100000</a> (terms n = 1..1000 from G. C. Greubel)
%F A185283 a(n) ~ c * sqrt(n), where c = 2*sqrt(3)/Pi. - _Amiram Eldar_, Dec 27 2024
%e A185283 a(3) = 2 because sigma(1) + sigma(2) + sigma(3) = 1+3+4 > 3.
%p A185283 b:= proc(n) option remember; `if`(n=0, 0,
%p A185283        numtheory[sigma](n)+b(n-1))
%p A185283     end:
%p A185283 a:= proc(n) option remember; local k; for k from
%p A185283       `if`(n=0, 0, a(n-1)) do if b(k)>=n then return k fi od
%p A185283     end:
%p A185283 seq(a(n), n=0..120);  # _Alois P. Heinz_, Sep 12 2019
%t A185283 a[n_] := (k = 1; While[ Total[ DivisorSigma[1, Range[k]]] < n, k++]; k); Table[ a[n], {n, 1, 90}]
%t A185283 Module[{nn=10,ad,th},ad={#[[1]],#[[2]]}&/@Partition[Accumulate[ DivisorSigma[ 1,Range[nn]]],2,1];th=Thread[{Range[2,nn],ad}];Join[ {0,1},Flatten[Table[#[[1]],#[[2,2]]-#[[2,1]]]&/@th]]] (* _Harvey P. Dale_, Aug 29 2020 *)
%Y A185283 Cf. A000203, A024916.
%K A185283 nonn
%O A185283 0,3
%A A185283 _Michel Lagneau_, Jan 21 2012
%E A185283 a(0)=0 prepended by _Alois P. Heinz_, Sep 12 2019