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A092530 a(0) = 0; for n > 0, a(n) = T(n) + k where T(n) is the n-th triangular number (A000217) and k (see A026741) is the smallest positive number such that a(n) is divisible by n.

%I A092530 #28 Sep 21 2024 15:24:23
%S A092530 0,2,4,9,12,20,24,35,40,54,60,77,84,104,112,135,144,170,180,209,220,
%T A092530 252,264,299,312,350,364,405,420,464,480,527,544,594,612,665,684,740,
%U A092530 760,819,840,902,924,989,1012,1080,1104,1175,1200,1274,1300,1377,1404,1484
%N A092530 a(0) = 0; for n > 0, a(n) = T(n) + k where T(n) is the n-th triangular number (A000217) and k (see A026741) is the smallest positive number such that a(n) is divisible by n.
%H A092530 Colin Barker, <a href="/A092530/b092530.txt">Table of n, a(n) for n = 0..1000</a>
%H A092530 A. W. Vyawahare and K. M. Purohit, <a href="https://core.ac.uk/download/pdf/30438744.pdf">Near pseudo Smarandache function</a>, Smarandache Notions, 14 (2004), 42-61.
%H A092530 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A092530 a(0) = 0, a(2n) = a(2n-1) + n, a(2n-1) = a(2n-2) + 3n-1. - _Amarnath Murthy_, Jul 04 2004
%F A092530 From _Colin Barker_, Feb 03 2019: (Start)
%F A092530 G.f.: x*(2 + 2*x + x^2 - x^3) / ((1 - x)^3*(1 + x)^2).
%F A092530 a(n) = (n*(2 + n)) / 2 for n even.
%F A092530 a(n) = (n*(3 + n)) / 2 for n odd.
%F A092530 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F A092530 (End)
%p A092530 seq(n*(1+ceil(n/2)), n=0..53); # _Zerinvary Lajos_ and _Klaus Brockhaus_, Apr 10 2007
%t A092530 {0}~Join~Array[Block[{k = 1}, While[GCD[#1, #2 + k] < #1, k++]; #2 + k] & @@ {#, (#^2 + #)/2} &, 53] (* or *)
%t A092530 CoefficientList[Series[x (2 + 2 x + x^2 - x^3)/((1 - x)^3*(1 + x)^2), {x, 0, 53}], x] (* _Michael De Vlieger_, Feb 03 2019 *)
%t A092530 LinearRecurrence[{1,2,-2,-1,1},{0,2,4,9,12},60] (* _Harvey P. Dale_, Sep 21 2024 *)
%o A092530 (PARI) for(n=0,53,print1(n*(1+ceil(n/2)),",")); \\ _Klaus Brockhaus_, Apr 10 2007
%o A092530 (PARI) concat(0, Vec(x*(2 + 2*x + x^2 - x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ _Colin Barker_, Feb 03 2019
%Y A092530 Equals A000217 + A026741.
%K A092530 nonn,easy
%O A092530 0,2
%A A092530 _N. J. A. Sloane_, Apr 08 2004