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A256188 In positive integers: replace k*(k+1)/2 with the first k numbers.

%I A256188 #21 Jun 04 2022 13:22:18
%S A256188 1,2,1,2,4,5,1,2,3,7,8,9,1,2,3,4,11,12,13,14,1,2,3,4,5,16,17,18,19,20,
%T A256188 1,2,3,4,5,6,22,23,24,25,26,27,1,2,3,4,5,6,7,29,30,31,32,33,34,35,1,2,
%U A256188 3,4,5,6,7,8,37,38,39,40,41,42,43,44,1,2,3,4
%N A256188 In positive integers: replace k*(k+1)/2 with the first k numbers.
%C A256188 a(A002061(n)) = 1;
%C A256188 a(A253169(n)) = n and a(m) != n for m < A253169(n);
%C A256188 a(A000537(n)) = A000217(n) and a(m) != A000217(n) for m < A000537(n);
%C A256188 see A004202 and A014132 for record values greater than 1 and where they occur: A014132(n) = a(A004202(n)).
%H A256188 Reinhard Zumkeller, <a href="/A256188/b256188.txt">Table of n, a(n) for n = 1..10000</a>
%e A256188 .  A000217 | 1,  3,      6,          10,                 15,       . . .
%e A256188 .  A000027 | _,2,___,4,5,_____,7,8,9,_______,11,12,13,14,_________,16,...
%e A256188 .  A002260 | 1,  1,2,    1,2,3,      1,2,3,4,            1,2,3,4,5,
%e A256188 .  --------+-------------------------------------------------------------
%e A256188 .     a(n) | 1,2,1,2,4,5,1,2,3,7,8,9,1,2,3,4,11,12,13,14,1,2,3,4,5,16,17,...
%t A256188 Table[If[OddQ[Sqrt[8n+1]],Range[(Sqrt[8n+1]-1)/2],n],{n,50}]//Flatten (* _Harvey P. Dale_, Jun 01 2019 *)
%o A256188 (Haskell)
%o A256188 a256188 n = a256188_list !! (n-1)
%o A256188 a256188_list = f 0 [1..] a002260_tabl where
%o A256188    f k xs (zs:zss) = us ++ zs ++ f (k + 1) vs zss
%o A256188                      where (us, v:vs) = splitAt k xs
%Y A256188 Cf. A255437, A000217, A014132, A002260, A000537, A004202, A014132, A002061, A255878 (first differences), A255879 (partial sums).
%K A256188 nonn
%O A256188 1,2
%A A256188 _Reinhard Zumkeller_, Mar 26 2015