A007606 Take 1, skip 2, take 3, etc. (Formerly M3241)

1, 4, 5, 6, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 45, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 137, 138

Offset

1,2

Comments

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.

a(A000290(n)) = A000384(n). - Reinhard Zumkeller, Feb 12 2011

A057211(a(n)) = 1. - Reinhard Zumkeller, Dec 30 2011

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

Union of nonzero terms of A000384 and A317304. - Omar E. Pol, Aug 29 2018

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

References

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Smarandache, Properties of Numbers, 1972.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.

Index entries for sequences generated by sieves

Formula

a(n) = n + m*(m+1) where m = floor(sqrt(n-1)). - Klaus Brockhaus, Mar 26 2004

a(n+1) = a(n) + if n=k^2 then 2*k+1 else 1; a(1) = 1. - Reinhard Zumkeller, May 13 2009

Example

From Omar E. Pol, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the odd numbers the sequence begins:
    1;
    4,   5,   6;
   11,  12,  13,  14,  15;
   22,  23,  24,  25,  26,  27,  28;
   37,  38,  39,  40,  41,  42,  43,  44,  45;
   56,  57,  58,  59,  60,  61,  62 , 63,  64,  65,  66;
   79,  80,  81,  82 , 83,  84,  85,  86,  87,  88,  89,  90,  91;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120;
...
Row sums give A005917.
Column 1 gives A084849.
Column 2 gives A096376, n >= 1.
Right border gives A000384, n >= 1.
(End)

Mathematica

Flatten[ Table[i, {j, 1, 17, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
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 *)
With[{nn=20}, Take[TakeList[Range[(nn(nn+1))/2], Range[nn]], {1, nn, 2}]]//Flatten (* Harvey P. Dale, Feb 10 2023 *)

Prog

(PARI) for(n=1, 66, m=sqrtint(n-1); print1(n+m*(m+1), ", "))
(Haskell)
a007606 n = a007606_list !! (n-1)
a007606_list = takeSkip 1 [1..] where
   takeSkip k xs = take k xs ++ takeSkip (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011

Crossrefs

Complement of A007607.

Cf. A007950, A007951, A007952, A048859, A004201.

Cf. A000384, A005917, A084849, A096376.

Sequence in context: A057916 A249853 A162415 * A284513 A047311 A224081

Adjacent sequences: A007603 A007604 A007605 * A007607 A007608 A007609

Keyword

nonn,tabf,nice,easy

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Status

approved