A004280 -id:A004280 - cp's OEIS frontend

A055398 Result of fourth stage of sieve of Eratosthenes (after eliminating multiples of 2, 3, 5, 7).

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 233, 239

Offset: 1

Henry Bottomley, May 15 2000

nonn

Essentially the same as A052424. - R. J. Mathar, Oct 13 2008

Ahmed Hamdy A. Diab, Sequence eliminating law (SEL) and the interval formulas of prime numbers, arXiv:2012.03052 [math.NT], 2020.
H. B. Meyer, Eratosthenes' sieve

Cf. A000040, A004280, A038179, A055396, A055399.

Join[{2,3,5,7},Select[Table[n,{n,2,500}],Mod[#,2]!=0&&Mod[#,3]!=0&&Mod[#,5]!=0&&Mod[#,7]!=0&]] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)

a(n+4) = 2 * floor((3 * floor((10 * floor((9 * floor((56 * floor((55 * floor((54 * floor((53 * floor((52 * floor((51 * floor((50 * floor((49 * n + 1) / 48) + 13) / 49) + 20) / 50) + 24) / 51) + 31) / 52) + 35) / 53) + 42) / 54) + 54) / 55) + 1) / 8) + 8) / 9) + 1) / 2) + 1 for n>=1; see (22) in Diab link. - Michel Marcus, Dec 14 2020

A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 7, 2, 11, 2, 9, 8, 9, 2, 17, 2, 17, 10, 13, 2, 23, 6, 15, 10, 23, 2, 29, 2, 17, 14, 19, 12, 35, 2, 21, 16, 37, 2, 41, 2, 35, 38, 25, 2, 47, 8, 47, 20, 41, 2, 53, 16, 51, 22, 31, 2, 59, 2, 33, 52, 33, 18, 65, 2, 53, 26, 67, 2, 71, 2, 39, 68, 59, 18, 77, 2, 77, 28, 43, 2, 83, 22, 45, 32, 79

Offset: 0

Alois P. Heinz, Dec 04 2018

Numbers where a(n) + A000010(n) != n + 1: A102467. - Robert G. Wilson v, Aug 23 2021

			a(6) = |{0,2,3,4,6}| = 5.
a(9) = |{0,3,6,9}| = 4.
a(10) = |{0,2,4,5,6,8,10}| = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Matt Baker, The Balanced Centrifuge Problem, Math Blog, 2018.
Holly Krieger and Brady Haran, The Centrifuge Problem, Numberphile video (2018)
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.
Robert G. Wilson v, Graph of the first 100001 terms.

Cf. A000040, A001248, A001748, A001750, A004280, A008588, A008864, A100484, A103306, A103314, A163300, A182986, A272470, A306275.

a:= proc(n) option remember; local f, b; f, b:=
       map(i-> i[1], ifactors(n)[2]),
       proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or f[i]<=m and b(m-f[i], i))
       end; forget(b); (t-> add(
      `if`(b(j, t) and b(n-j, t), 1, 0), j=0..n))(nops(f))
    end:
seq(a(n), n=0..100);

$RecursionLimit = 4096;
a[1] = 0;
a[n_] := a[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]];
     b[m_, i_] := b[m, i] = m == 0 || i > 0 &&
     (b[m, i - 1] || f[[i]] <= m && b[m - f[[i]], i]);
     With[{t = Length[f]}, Sum[
     If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]]];
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz, corrected and updated Aug 07 2021 *)
f[n_] := Block[{c = 2, k = 2, p = First@# & /@ FactorInteger@ n}, While[k < n, If[ IntegerPartitions[k, All, p, 1] != {} && IntegerPartitions[n - k, All, p, 1] != {}, c++]; k++]; c]; f[0] = 1; f[1] = 0; Array[f, 75] (* Robert G. Wilson v, Aug 22 2021 *)

a(n) = |{ k : k and n-k can be written as a sum of prime factors of n }|.
a(n) = 2 <=> n is prime (A000040).
a(n) >= n-1 <=> n in {1,2,3,4} union { A008588 }.
a(n) = (n+4)/2 <=> n in { A100484 } minus { 4 }.
a(n) = (n+9)/3 <=> n in { A001748 } minus { 9 }.
a(n) = (n+25)/5 <=> n in { A001750 } minus { 25 }.
a(n) = (n+49)/7 <=> n in { A272470 } minus { 49 }.
a(n^2) = n+1 <=> n = 0 or n is prime <=> n in { A182986 }.
a(A001248(n)) = A008864(n).
a(n) is odd <=> n in { A163300 }.
a(n) is even <=> n in { A004280 }.

A054403 Result of third stage of sieve of Eratosthenes (after eliminating multiples of 2, 3 and 5).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211

Offset: 1

Henry Bottomley, May 15 2000

nonn

Ahmed Hamdy A. Diab, Sequence eliminating law (SEL) and the interval formulas of prime numbers, arXiv:2012.03052 [math.NT], 2020.
H. B. Meyer, Eratosthenes' sieve
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A000040, A004280, A038179, A055396, A055398, A055399.

Join[{2,3,5},Select[Table[n,{n,2,300}],Mod[#,2]!=0&&Mod[#,3]!=0&&Mod[#,5]!=0&]] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011*)

2, 3, 5 and numbers 30k +/- 1, 7, 11, or 13.

a(n+3) = 2*floor((3*floor((10*floor((9*n+1)/8)+8)/9)+1)/2)+1 for n>=1; see (19) in Diab link. - Michel Marcus, Dec 14 2020

A168007 Jumping divisor sequence (see Comments lines for definition).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 9, 12, 11, 22, 21, 24, 23, 46, 45, 48, 47, 94, 93, 96, 95, 100, 99, 102, 101, 202, 201, 204, 203, 210, 209, 220, 219, 222, 221, 234, 233, 466, 465, 468, 467, 934, 933, 936, 935, 940, 939, 942, 941, 1882, 1881, 1884, 1883, 1890, 1889, 3778, 3777, 3780, 3779, 7558, 7557, 7560, 7559

Offset: 1

Omar E. Pol, Nov 19 2009

Consider the diagram with overlapping periodic curves that appears in the Links section (figure 2). The number of curves that contain the point [n,0] equals the number of divisors of n. The curve of diameter d represents the divisor d of n. Now consider only the lower part of the diagram (figure 3). Starting from point [1,0] we continue our journey walking along the semicircumference with smallest diameter not used previously (see the illustration of initial terms, figure 1). The sequence is formed by the values of n where the trajectory intercepts the x axis. - Omar E. Pol, Jan 14 2019

Jinyuan Wang, Table of n, a(n) for n = 1..1000
Omar E. Pol, Illustration of initial terms (Fig. 1)
Omar E. Pol, Periodic curves and tau(n) (Fig. 2)
Omar E. Pol, Periodic curves and tau(n), lower part upside down (Fig. 3)

Cf. A000005, A002808, A020639, A004280, A168008, A168009.

lista(nn) = {my(v=vector(nn, i, if(i<4, 2^i/2))); for(n=4, nn, if(v[n-1]%2, v[n]=v[n-1] + factor(v[n-1])[1, 1], v[n]=v[n-1] - 1)); v; } \\ Jinyuan Wang, Mar 14 2020

from itertools import count, islice
from sympy import primefactors
def A168007_gen(): # generator of terms
    yield (a := 1)
    for n in count(2):
        yield (a:=a+(min(primefactors(a),default=1) if a&1 or a==2 else -1))
A168007_list = list(islice(A168007_gen(),20)) # Chai Wah Wu, Mar 14 2023

a(1) = 1; if a(n) is an even composite number then a(n+1) = a(n) - 1; otherwise a(n+1) = a(n) + A020639(a(n)). - Omar E. Pol, Jan 13 2019

More terms from Omar E. Pol, Jan 12 2019

A370952 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod 2*n-1).

Original entry on oeis.org

1, 4, 9, 2, 11, 22, 35, 5, 39, 20, 41, 18, 43, 16, 45, 14, 47, 12, 49, 10, 51, 8, 53, 6, 55, 106, 159, 104, 161, 102, 163, 37, 167, 33, 171, 29, 175, 25, 179, 21, 183, 17, 187, 13, 191, 100, 7, 197, 3, 201, 302, 96, 306, 92, 310, 88, 314, 84, 318, 80, 322, 76, 326, 72, 330, 68, 334, 64, 338, 60, 342, 56, 346, 52, 350

Offset: 1

N. J. A. Sloane, Mar 07 2024

nonn

Very similar to A368382, except that there the differences are controlled by the sequence {2,3,5,7,9,11,13,15,...} (A004280), whereas here they are controlled by {3,5,7,9,11,13,15,...}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004280, A368382, A370975, A372052 (where n appears).

from itertools import count, islice
def A370952_gen(): # generator of terms
    a, aset = 1, {0,1}
    for p in count(3,2):
        yield a
        for b in count(a%p,p):
            if b not in aset:
                aset.add(b)
                a = b
                break
A370952_list = list(islice(A370952_gen(),20)) # Chai Wah Wu, Apr 17 2024

A303600 a(1)=2 and a(2)=3, then a(n+1) is the smallest integer larger than a(n) that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 11, 16, 22, 24, 28, 29, 30, 37, 42, 50, 55, 56, 70, 73, 76, 82, 89, 95, 101, 102, 115, 128, 133, 135, 136, 141, 142, 153, 160, 161, 168, 174, 181, 195, 199, 200, 214, 221, 227, 233, 247, 252, 265, 266, 267, 273, 280, 285, 286, 325, 331, 332, 338

Offset: 1

Michel Marcus, Apr 26 2018

nonn

David A. Corneth, Table of n, a(n) for n = 1..10000 (Terms 1..100 from David Consiglio, Jr.)
David A. Corneth, PARI-prog
Borys Kuca, Structures in Additive Sequences, arXiv:1804.09594 [math.NT], 2018. See V(2,3).

Cf. A004280 (with first terms 1 and 2).

Nest[Append[#, Function[{m, s}, First@ SelectFirst[Tally[s], And[First@ # > m, Last@ # < 3] &]] @@ {Max@ #, Sort[Total /@ Tuples[#, {2}]]}] &, {2, 3}, 57] (* Michael De Vlieger, Apr 27 2018 *)

\\ See PARI link \\ David A. Corneth, Apr 27 2018

terms = [2,3]
while len(terms) < 100:
    options = []
    for x in range(len(terms)):
        for y in range(x,len(terms)):
            options.append(terms[x]+terms[y])
    for y in sorted(options):
        if options.count(y) == 1 and y > max(terms):
            terms.append(y)
            break
print(terms)
# David Consiglio, Jr., Apr 18 2018

from collections import Counter
def A303600_list(num_terms):
    terms = [2, 3]
    sum_counts = Counter([4, 5, 6])
    next_term = 3
    for n in range(2, num_terms):
        next_term += 1
        while sum_counts[next_term] != 1: next_term += 1
        terms.append(next_term)
        for term in terms: sum_counts[term + next_term] += 1
    return terms
print(A303600_list(59))
# David Radcliffe, Jun 19 2025

More terms from David Consiglio, Jr., Apr 26 2018

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 24 2018

Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves.

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 1, 1, ...
  1, 1, 3, 2, 2, 2, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  ...

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on this sequence and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

Cf. A300453, A300454, A316989, A321126, A321127.

b[n_] = If[n == 0 || n == 2, 1, 0];
T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten

b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
create_list(T(k, n - k), n, 0, 12, k, 0, n);

T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n).
T(n,1) = A300453(n+1,A321126(n,1)).
T(n,2) = A300454(n,A321126(n,2)).
T(n,n) = A321127(n,A004280(n+1)).
G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).
E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.

A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 4, 5, 6, 7, 6, 5, 5, 5, 6, 7, 8, 7, 6, 6, 6, 6, 7, 8, 9, 8, 7, 7, 7, 7, 7, 8, 9, 10, 9, 8, 8, 8, 8, 8, 8, 9, 10, 11, 10, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13

Offset: 0

Franck Maminirina Ramaharo, Nov 21 2018

T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial (A,B,d) for the two-bridge knot with Conway's notation C(n,k). Hence, T(n,k) is the maximum number of Jordan curves that are obtained by splitting the crossings of such knot diagram.

			Square array begins:
    1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    2,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    3,  3,  3,  4,  5,  6,  7,  8,  9, 10, ...
    4,  4,  4,  5,  6,  7,  8,  9, 10, 11, ...
    5,  5,  5,  6,  7,  8,  9, 10, 11, 12, ...
    6,  6,  6,  7,  8,  9, 10, 11, 12, 13, ...
    7,  7,  7,  8,  9, 10, 11, 12, 13, 14, ...
    8,  8,  8,  9, 10, 11, 12, 13, 14, 15, ...
    9,  9,  9, 10, 11, 12, 13, 14, 15, 16, ...
   10, 10, 10, 11, 12, 13, 14, 15, 16, 17, ...
  ...

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on sequence A321125 and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

T(n,1) = degree of the (n+1)-th row polynomial in A300453.
T(n,k) = degree of the n-th row polynomials in A300454 and A321127, k = 2,n, respectively.
Cf. A321125, A316989.
Cf. A003056, A003983, A051125.

Table[Max[k + 1, n - 1, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten

create_list(max(k + 1, n - 1, n - k + 1), n, 0, 10, k, 0, n);

T(n,k) = T(k,n).
T(n,k) = A051125(n+1,k+1) for 0 <= k <= 2, n >= 0, and T(n,k) =  A051125(n+1,k+1) + A003983(n-2,k-2) for k >= 3, n >= 3.
T(n,n) = A004280(n+1).
G.f.: (1 - (2*x - x^2)*y + (x - 2*x^2 + x^3)*y^2 + (x^2 - x^3)*y^3)/(((1 - x)*(1 - y))^2).

A382850 a(n) = least k such that binomial(n, k) > binomial(n - 1, h) for 0 <= h <= n - 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31

Offset: 2

Clark Kimberling, Apr 07 2025

nonn

All runs appear to have length 2 or 3, with no two consecutive runs of length 3, except for the first two (1, 1, 1; 2, 2, 2). The length of the k-th run of consecutive length-2 runs seems to be close to 0.722*k, but varies irregularly: 1, 2, 2, 4, 3, 5, 5, 7, 6, 8, 8, 9, 9, ... . - Pontus von Brömssen, Apr 15 2025

(Runlength sequence of the runlength sequence) = (2,1,1,2,1,2,1,4,1,3,1,5,...) appears to have 1 in positions given by A004280(n) for n >= 2. - Clark Kimberling, Apr 15 2025

			The least k such that binomial(5,k) > binomial(4,h) for 0<=h<=4 is 3, since 10 > max{1,4,6}.

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A001405, A007318, A382851.

z = 40; c[n_, k_] := Binomial[n, k];
t[n_] := Table[c[n, k], {k, 0, n}];
a[n_] := Select[Range[z], c[n, #] > c[n - 1, Floor[(n - 1)/2]] &, 1];
Flatten[Table[a[n], {n, 1, 3 z}]]  (* A382850 *)
Flatten[Table[c[n, a[n]], {n, 1, z}]]  (* A382851 *)

row(n) = vector(n+1, k, binomial(n,k-1));
a(n) = my(val = vecmax(row(n-1)), w = row(n)); for (i=1, #w, if (w[i] > val, return(i-1));); \\ Michel Marcus, Apr 14 2025

from itertools import count
def A382850_generator():
    k = C0 = C = 1
    for n in count(2):
        C0 = 2*C0 if n%2 == 1 else (n-1)*C0//(n//2) # C(n-1,floor((n-1)/2))
        C = C*n//(n-k) # C(n,k)
        if C <= C0:
            C = C*(n-k)//(k+1)
            k += 1
        yield k # Pontus von Brömssen, Apr 15 2025

a(n) - a(n-1) is either 0 or 1. - Pontus von Brömssen, Apr 15 2025

Apparently, the lower and upper limits of f(n) = a(n) - n/2 + sqrt(n*log(2)/2) are 0 and 1, respectively, with f(n) < 1 for all n, but with the minimum -0.0144167... of f attained at n = 184. - Pontus von Brömssen, Apr 16 2025

A216240 Composite numbers arising in Eratosthenes sieve with removing the multiples of every other remaining numbers after 2 (see comment).

Original entry on oeis.org

9, 21, 33, 49, 51, 77, 87, 119, 121, 123, 141, 177, 187, 201, 203, 219, 237, 287, 289, 291, 309, 319, 327, 329, 357, 393, 413, 417, 447, 451, 469, 471, 493, 501, 511, 517, 543, 553, 573, 591, 633, 649, 669, 679, 687, 697, 721, 723, 737, 763, 771, 799, 803, 807

Offset: 1

Vladimir Shevelev, Mar 14 2013

nonn

We remove even numbers except for 2. The first two remaining numbers are 3,5. Further we remove all remaining numbers multiple of 5,except for 5. The following two remaining numbers are 7,9. Now we remove all remaining numbers multiple of 9, except for 9, etc. The sequence lists the remaining composite numbers.

Conjecture. There exists x_0 such that for every x>=x_0, the number of a(n)<=x is more than pi(x).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A003309, A038179, A004280, A054103, A055398, A055399, A066680, A083140, A145592, A152021, A152114.

Module[{a=Insert[Range[1,1000,2], 2, 2], k=4}, While[Length[a] >= 2k, a = Flatten[{Take[a,k], Select[Take[a,-Length[a]+k], Mod[#,a[[k]]] != 0 &]}]; k+=2]; Rest[Select[a,!PrimeQ[#]&]]] (* Peter J. C. Moses, Mar 27 2013 *)

cp's OEIS Frontend

A055398 Result of fourth stage of sieve of Eratosthenes (after eliminating multiples of 2, 3, 5, 7).

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A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

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Maple

Mathematica

Formula

A054403 Result of third stage of sieve of Eratosthenes (after eliminating multiples of 2, 3 and 5).

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Mathematica

Formula

A168007 Jumping divisor sequence (see Comments lines for definition).

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PARI

Python

Formula

Extensions

A370952 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod 2*n-1).

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Python

A303600 a(1)=2 and a(2)=3, then a(n+1) is the smallest integer larger than a(n) that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way.

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Mathematica

PARI

Python

Python

Extensions

A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

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Mathematica

Maxima

Formula

A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

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A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).