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%I M3241 #51 Feb 10 2023 19:19:10
%S 1,4,5,6,11,12,13,14,15,22,23,24,25,26,27,28,37,38,39,40,41,42,43,44,
%T 45,56,57,58,59,60,61,62,63,64,65,66,79,80,81,82,83,84,85,86,87,88,89,
%U 90,91,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,137,138
%N Take 1, skip 2, take 3, etc.
%C List the natural numbers: 1, 2, 3, 4, 5, 6, 7, ... . Keep the first number (1), delete the next two numbers (2, 3), keep the next three numbers (4, 5, 6), delete the next four numbers (7, 8, 9, 10) and so on.
%C a(A000290(n)) = A000384(n). - _Reinhard Zumkeller_, Feb 12 2011
%C A057211(a(n)) = 1. - _Reinhard Zumkeller_, Dec 30 2011
%C Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central valley. (Cf. A237593.) - _Omar E. Pol_, Aug 28 2018
%C Union of nonzero terms of A000384 and A317304. - _Omar E. Pol_, Aug 29 2018
%C The values of k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class r' mod m' (with r' in {1,...,m'}) iff m<m' or r<r'. Cf. A360418. - _James Propp_, Feb 10 2023
%D C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
%D R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D F. Smarandache, Properties of Numbers, 1972.
%H Reinhard Zumkeller, <a href="/A007606/b007606.txt">Table of n, a(n) for n = 1..10000</a>
%H C. Dumitrescu & V. Seleacu, editors, <a href="http://www.gallup.unm.edu/~smarandache/SNAQINT.txt">Some Notions and Questions in Number Theory, Vol. I</a>.
%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F a(n) = n + m*(m+1) where m = floor(sqrt(n-1)). - _Klaus Brockhaus_, Mar 26 2004
%F a(n+1) = a(n) + if n=k^2 then 2*k+1 else 1; a(1) = 1. - _Reinhard Zumkeller_, May 13 2009
%e From _Omar E. Pol_, Aug 29 2018: (Start)
%e Written as an irregular triangle in which the row lengths are the odd numbers the sequence begins:
%e 1;
%e 4, 5, 6;
%e 11, 12, 13, 14, 15;
%e 22, 23, 24, 25, 26, 27, 28;
%e 37, 38, 39, 40, 41, 42, 43, 44, 45;
%e 56, 57, 58, 59, 60, 61, 62 , 63, 64, 65, 66;
%e 79, 80, 81, 82 , 83, 84, 85, 86, 87, 88, 89, 90, 91;
%e 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120;
%e ...
%e Row sums give A005917.
%e Column 1 gives A084849.
%e Column 2 gives A096376, n >= 1.
%e Right border gives A000384, n >= 1.
%e (End)
%t Flatten[ Table[i, {j, 1, 17, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* _Robert G. Wilson v_, Mar 11 2004 *)
%t Join[{1},Flatten[With[{nn=20},Range[#[[1]],Total[#]]&/@Take[Thread[ {Accumulate[ Range[nn]]+1,Range[nn]}],{2,-1,2}]]]] (* _Harvey P. Dale_, Jun 23 2013 *)
%t With[{nn=20},Take[TakeList[Range[(nn(nn+1))/2],Range[nn]],{1,nn,2}]]//Flatten (* _Harvey P. Dale_, Feb 10 2023 *)
%o (PARI) for(n=1,66,m=sqrtint(n-1);print1(n+m*(m+1),","))
%o (Haskell)
%o a007606 n = a007606_list !! (n-1)
%o a007606_list = takeSkip 1 [1..] where
%o takeSkip k xs = take k xs ++ takeSkip (k + 2) (drop (2*k + 1) xs)
%o -- _Reinhard Zumkeller_, Feb 12 2011
%Y Complement of A007607.
%Y Cf. A007950, A007951, A007952, A048859, A004201.
%Y Cf. A000384, A005917, A084849, A096376.
%K nonn,tabf,nice,easy
%O 1,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_
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