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A316571 a(1) = 1; for n > 1: a(n) = smallest number such that (Sum_{k=1..n} a(k)) is divisible by n - 1.

%I A316571 #32 Jun 02 2025 15:22:05
%S A316571 1,2,3,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,
%T A316571 48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,
%U A316571 94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124
%N A316571 a(1) = 1; for n > 1: a(n) = smallest number such that (Sum_{k=1..n} a(k)) is divisible by n - 1.
%C A316571 This is the lexicographically earliest increasing sequence such that n-1 divides the sum of the first n terms.
%C A316571 Sequence b(n) of the sums of the first n+1 terms, Sum_{k=1..n+1} a(k): 3, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, ...
%C A316571 Sequence c(n) of quotients when a(n) is calculated, (Sum_{k=1..n+1} a(k) ) / n is 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
%H A316571 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).
%F A316571 a(1)=1, a(2)=2, a(3)=3, a(n)=2(n-1) for n >= 4.
%F A316571 a(n) = A005843(n-1) for n >= 4. - _Antti Karttunen_, Sep 16 2018
%F A316571 G.f.: (1 + 2*x^3 - x^4)/(-1 + x)^2. - _Stefano Spezia_, Sep 16 2018
%F A316571 a(n) = A020739(n-4) = 2*(n - 4) + 6 for n >= 4. - _Georg Fischer_, Jan 19 2019
%e A316571 a(1) = 1 because 1 divides the sum of the first 2 (i.e., 1 + 1) terms (a(1) + a(2)) for whatever term a(2) > a(1).
%e A316571 a(2) = 2 because 2 is the smallest number > a(1) and 2 divides the sum of the first 3 (i.e., 2 + 1) terms (a(1) + a(2) + a(3)) for whatever term a(3) > a(2) such that 2 divides the sum a(1) + a(2) + a(3); the smallest number > a(2) with this property for a(3) is 3.
%e A316571 a(3) = 3.
%e A316571 a(4) = 6 because 6 is the smallest number > a(3) such that 3 divides the sum of the first 4 (i.e., 3 + 1) terms.
%e A316571 a(5) = 8 because 8 is the smallest number > a(4) such that 4 divides the sum of the first 5 (i.e., 4 + 1) terms.
%t A316571 CoefficientList[Series[(1 + 2 x^3 - x^4)/(-1 + x)^2 , {x, 0, 50}], x] (* _Stefano Spezia_, Sep 16 2018 *)
%t A316571 Join[{1, 2, 3}, LinearRecurrence[{2, -1}, {6, 8}, 60]] (* _Jean-François Alcover_, Dec 30 2018 *)
%Y A316571 Cf. A005843, A020739, A316156.
%K A316571 nonn
%O A316571 1,2
%A A316571 _Jaroslav Krizek_, Aug 20 2018
%E A316571 Definition clarified by _Georg Fischer_ and _Alois P. Heinz_, Jan 19 2019