A007951 -id:A007951 - cp's OEIS frontend

A139387 First gap of length at least n in A007951, lower end.

2, 11, 133, 497, 4963, 34486, 68978, 14877929, 6186632564, 6186632564

Offset: 1

Klaus Brockhaus, Apr 17 2008

a(11) > 10^11. - Donovan Johnson

			First few terms of A007951 are 1,2,4,5,7,8,10,11,14,16, ... . The first gap of length 1 is between 2 and 4, so a(1) = 2; the first gap of length 2 is between 11 and 14, so a(2) = 11.

Cf. A007951, A138845, A138900, A139386.

a(8)-a(10) from Donovan Johnson, Dec 27 2010

A007606 Take 1, skip 2, take 3, etc.

Original entry on oeis.org

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

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

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 mA360418. - James Propp, Feb 10 2023

			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)

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.

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

Complement of A007607.
Cf. A007950, A007951, A007952, A048859, A004201.
Cf. A000384, A005917, A084849, A096376.

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

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 *)

for(n=1,66,m=sqrtint(n-1);print1(n+m*(m+1),","))

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

A007950 Binary sieve: delete every 2nd number, then every 4th, 8th, etc.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 17, 21, 25, 27, 29, 33, 35, 37, 43, 49, 51, 53, 57, 59, 65, 67, 69, 73, 75, 77, 81, 85, 89, 91, 97, 101, 107, 109, 113, 115, 117, 121, 123, 129, 131, 133, 137, 139, 145, 149, 153, 155, 157, 161, 163, 165, 171, 173, 177, 179, 181, 185, 187, 195, 197

Offset: 1

R. Muller

From Charles T. Le (charlestle(AT)yahoo.com), Mar 22 2004: (Start)
This sequence and A007951 are particular cases of the Smarandache n-ary sequence sieve (for n=2 and respectively n=3).
Definition of Smarandache n-ary sieve (n >= 2): Starting to count on the natural numbers set at any step from 1: - delete every n-th numbers; - delete, from the remaining numbers, every (n^2)-th numbers; ... and so on: delete, from the remaining ones, every (n^k)-th numbers, k = 1, 2, 3, ... .
Conjectures: there are infinitely many primes that belong to this sequence; also infinitely many composite numbers.
Smarandache general-sequence sieve: Let u_i > 1, for i = 1, 2, 3, ..., be a strictly increasing positive integer sequence. Then from the natural numbers: - keep one number among 1, 2, 3, ..., u_1 - 1 and delete every u_1 -th numbers; - keep one number among the next u_2 - 1 remaining numbers and delete every u_2 -th numbers; ... and so on, for step k (k >= 1): - keep one number among the next u_k - 1 remaining numbers and delete every u_k -th numbers; ... (End)
Certainly this sequence contains infinitely many composite numbers, as it has finite density A048651, while the primes have zero density. - Franklin T. Adams-Watters, Feb 25 2011

F. Smarandache, Properties of Numbers, 1972.

T. D. Noe, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 95.
Index entries for sequences generated by sieves

Cf. A007951, A000959, A048651.

t = Range@200; f[n_] := Block[{k = 2^n}, t = Delete[t, Table[{k}, {k, k, Length@t, k}]]]; Do[ f@n, {n, 6}]; t (* Robert G. Wilson v, Sep 14 2006 *)

More terms from Robert G. Wilson v, Sep 14 2006

A048859 A sieve: keep the first 2 numbers, delete the next 3 numbers; keep the next 3 numbers, delete the next 4 numbers; keep the next 4 numbers, delete the next 5 numbers; and so on. In other words, keep the next k numbers and delete the next k+1 numbers, for k = 2, 3, ...

Original entry on oeis.org

1, 2, 6, 7, 8, 13, 14, 15, 16, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 38, 46, 47, 48, 49, 50, 51, 52, 61, 62, 63, 64, 65, 66, 67, 68, 78, 79, 80, 81, 82, 83, 84, 85, 86, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

nonn

			List the natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
Keep the first two numbers 1, 2 and delete the next three numbers 3, 4, 5.
Keep the next three numbers 6, 7, 8 and delete the next four numbers 9, 10, 11, 12. And so on.

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
M. Le, On the Smarandache n-ary Sieve, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 146-147.
F. Smarandache, Properties of Numbers, 1972.

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

Cf. A007950, A007951, A007952.

Cf. A064801.

a048859 n = a048859_list !! (n-1)
a048859_list = f 2 [1..] where
   f k xs = us ++ f (k + 1) (drop (k + 1) vs)
            where (us, vs) = splitAt k xs
-- Reinhard Zumkeller, May 16 2014

ss[n_]:=Module[{c=n^2+4n+1},Range[c,c+n+1]]; Flatten[Array[ss,10,0]] (* Harvey P. Dale, Sep 10 2014 *)

Corrected and revised by the author, Mar 24 2004
More terms from Bernardo Boncompagni Jul 27 2004
Offset changed by Reinhard Zumkeller, May 16 2014

A092418 A sieve: starting with the sequence of positive integers, delete every 4th number, then delete every 16th number from the remaining sequence, then delete every 64th number, etc. Sequence gives the remaining numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com), Mar 22 2004

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/4^k) = 0.688537... (A100221). - Amiram Eldar, Mar 21 2021

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
Florentin Smarandache, Properties of Numbers, 1972.

Amiram Eldar, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, item 31.

Cf. A007950, A007951, A100221.

A = 1:200; A(4:4:end) = 0; A = A(find(A)); A(16:16:end) = 0; A = A(find(A)); A(64:64:end) = 0; A = A(find(A))
% David Wasserman, Apr 28 2004

Edited by David Wasserman, Apr 28 2004

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A139387 First gap of length at least n in A007951, lower end.

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A007606 Take 1, skip 2, take 3, etc.

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A007950 Binary sieve: delete every 2nd number, then every 4th, 8th, etc.

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A048859 A sieve: keep the first 2 numbers, delete the next 3 numbers; keep the next 3 numbers, delete the next 4 numbers; keep the next 4 numbers, delete the next 5 numbers; and so on. In other words, keep the next k numbers and delete the next k+1 numbers, for k = 2, 3, ...

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A092418 A sieve: starting with the sequence of positive integers, delete every 4th number, then delete every 16th number from the remaining sequence, then delete every 64th number, etc. Sequence gives the remaining numbers.

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