A099862 -id:A099862 - cp's OEIS frontend

A099861 a(n) = (2*n-1)-st composite number: a bisection of A002808.

4, 8, 10, 14, 16, 20, 22, 25, 27, 30, 33, 35, 38, 40, 44, 46, 49, 51, 54, 56, 58, 62, 64, 66, 69, 72, 75, 77, 80, 82, 85, 87, 90, 92, 94, 96, 99, 102, 105, 108, 111, 114, 116, 118, 120, 122, 124, 126, 129, 132, 134, 136, 140, 142, 144, 146, 148, 152, 154, 156, 159, 161

Offset: 1

N. J. A. Sloane, Nov 19 2004

			a(1) = 4 is the first composite number.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A002808, A099862, A175228.

b:=proc(n) if isprime(n)=true then else n fi end: B:=[seq(b(n),n=2..250)]: seq(B[2*m-1],m=1..75); # Emeric Deutsch, Dec 09 2004

Partition[Select[Range[200], CompositeQ], 2][[All, 1]] (* Jean-François Alcover, Mar 22 2023 *)

from sympy import composite
def A099861(n): return composite((n<<1)-1) # Chai Wah Wu, Nov 14 2024

a(n) = A175228(n+1). - A.H.M. Smeets, Aug 19 2019

More terms from Emeric Deutsch, Dec 09 2004

A175227 Sequence of the remaining nonprimes out of sieve of type: {step 1: starting with the sequence of positive integers A000027; step 2: delete every prime number A000040; step 3: delete every prime(1)-th number greater than the prime(1) from the remaining sequence; step 4: delete every prime(2)-th number greater than the prime(2) from the remaining sequence; etc. for prime(k)-th numbers for k = 3, 4, 5, ...}.

Original entry on oeis.org

1, 4, 8, 14, 16, 22, 30, 33, 40, 49, 54, 56, 62, 77, 80, 92, 94, 99, 111, 116, 118, 132, 144, 152, 154, 166, 174, 182

Offset: 1

Jaroslav Krizek, Mar 07 2010

nonn

Sequence of step 1: A000027, sequences of step 2: delete A000040 = primes, remaining A018252 = nonprimes, sequences of step 3: delete A099862 = {6, 9, 12, 15, 18, 21, 24, 26, 28, 32, ...}, remaining A175228 = {1, 4, 8, 10, 14, 16, 20, 22, 25, 27, ...}, sequences of step 4: delete A175229 = {10, 20, 27, 35, 44, 51, 58, 66, 75, 82, ...}, remaining A175230 = {1, 4, 8, 14, 16, 22, 25, 30, 33, 38, ...}, sequences of step 5: delete sequence = {25, 46, 64, 85, 102, 122, 140, 159, 176, 196, ...}, remaining sequence = {1, 4, 8, 14, 16, 22, 30, 33, 38, 40, ...}, ...

A377898 A121053 sorted into increasing order, or, equivalently, the indices of prime terms in A121053.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 27, 29, 30, 31, 33, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 51, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 69, 71, 72, 73, 75, 77, 79, 80, 82, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 102, 103, 105, 107, 108, 109, 111, 113, 114, 116, 118, 120, 122, 124, 126, 127, 129, 131, 132, 134, 136, 137, 139

Offset: 1

N. J. A. Sloane, Nov 14 2024

nonn

That the two definitions produce the same values is a consequence of the definition of A121053.

Cf. A121053.

Complement of A099862.

A379054 a(n) = composite(2n+2) - composite(2n).

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 4, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 2, 4, 2, 4, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3

Offset: 1

N. J. A. Sloane, Dec 18 2024

nonn

A121053 is divided into blocks, and (ignoring the initial terms of A121053) these are the lengths of the blocks.

Paolo Xausa, Table of n, a(n) for n = 1..10000

First differences of A099862.

Cf. A002808, A099861, A121053, A379055.

Differences[Select[Range[300], CompositeQ][[2 ;; ;; 2]]] (* Paolo Xausa, Dec 18 2024 *)

A354430 First diagonal of an array, generated from the sequence of the nonprimes.

Original entry on oeis.org

1, 7, 22, 58, 142, 334, 766, 1726, 3837, 8435, 18364, 39646, 84986, 181117, 384160, 811676, 1709425, 3590213, 7522354, 15728427, 32827027, 68405533, 142344708, 295824870, 614046159, 1273068141, 2636250146, 5452584131, 11264148401, 23242423457, 47903544728

Offset: 1

Tamas Sandor Nagy, May 27 2022

Mirroring the idea in A048457, here with nonprimes, and including 1 of the first generation.

We write down the sequence of the nonprimes 1, 4, 6, ... in the first row of the array. Nonprime(k) + nonprime(k+2) will generate the second row. Thereafter we generate the further rows in a similar manner. The leftmost diagonal gives the sequence.

			1    4    6    8    9   10   12   14   15   16   18   20   21 ...
     7   12   15   18   21   24   27   30   33   36   39 ...
         22   30   36   42   48   54   60   66   72 ...
              58   72   84   96  108  120  132 ...
                  142  168  192  216  240 ...
                       334  384  432 ...
                            766 ...

Michael S. Branicky, Table of n, a(n) for n = 1..3311

Cf. A001787, A018252, A048457, A048448, A099862.

from sympy import composite
from functools import lru_cache
@lru_cache(maxsize=None)
def T(r, k):
    if r == 1: return 1 if k == 1 else composite(k-1)
    return T(r-1, k) + T(r-1, k+2)
def a(n): return T(n, 1)
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, May 28 2022

a(8) and beyond from Michael S. Branicky, May 28 2022

A377900 After A121053(n) has been found, a(n) is the smallest candidate for A121053(n+1) that has not been eliminated.

Original entry on oeis.org

1, 1, 1, 6, 6, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 21, 21, 21, 24, 24, 24, 26, 26, 28, 28, 32, 32, 32, 32, 34, 34, 36, 36, 39, 39, 39, 42, 42, 42, 45, 45, 45, 48, 48, 48, 50, 50, 52, 52, 55, 55, 55, 57, 57, 60, 60, 60, 63, 63, 63, 65, 65, 68, 68, 68, 70

Offset: 1

N. J. A. Sloane, Nov 14 2024

nonn

			After a(8) = 9, and A121053(9) = 10 has been determined, the smallest prime not yet used is 17 and the smallest composite not yet used or eliminated is 12 (10 is now eliminated because the terms of A121053 must be distinct), so a(9) = 12.

Michael S. Branicky, Table of n, a(n) for n = 1..20000

Cf. A099862, A121053, A377898.

lista(nn) = my(c=4, t=0); print1("1, 1, 1"); forcomposite(k=4, nn, if(t%2, for(n=c, k-1, print1(", ", k)); c=k); t++); \\ Jinyuan Wang, Nov 29 2024

from sympy import isprime
from itertools import count, islice
def nextcomposite(n): return next(k for k in count(n+1) if not isprime(k))
def agen(): # generator of terms
    yield from [1, 1, 1]
    c, c2 = 4, 6
    for n in count(4):
        if n == c2: c, c2 = c2, nextcomposite(nextcomposite(c2))
        yield c2
print(list(islice(agen(), 70))) # Michael S. Branicky, Nov 29 2024

a(n) = A099862(k+1) for A099862(k) <= n < A099862(k+1). - Jinyuan Wang, Nov 29 2024

More terms from Jinyuan Wang, Nov 29 2024

cp's OEIS Frontend

A099861 a(n) = (2*n-1)-st composite number: a bisection of A002808.

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A377898 A121053 sorted into increasing order, or, equivalently, the indices of prime terms in A121053.

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A379054 a(n) = composite(2*n+2) - composite(2*n).

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Mathematica

A354430 First diagonal of an array, generated from the sequence of the nonprimes.

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A377900 After A121053(n) has been found, a(n) is the smallest candidate for A121053(n+1) that has not been eliminated.

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A379054 a(n) = composite(2n+2) - composite(2n).