A129131 -id:A129131 - cp's OEIS frontend

A129146 a(n) = n-th odd prime minus n-th odd composite number.

-6, -10, -14, -14, -14, -16, -16, -16, -16, -18, -14, -14, -14, -16, -12, -10, -14, -10, -10, -12, -8, -8, -4, 2, 2, -2, -4, -6, -4, 8, 10, 14, 14, 20, 18, 22, 22, 24, 28, 32, 28, 36, 34, 36, 34, 42, 52, 52, 52, 50, 54, 54, 62, 62, 62, 66, 66, 70, 72, 70, 78, 90, 92, 92, 92, 100, 102, 110, 106, 108, 112, 118, 120, 124, 124, 128

Offset: 1

Zak Seidov, Apr 01 2007

For small n's, a(n) is negative, while for large n's, a(n) is positive.

Cf. A065091, A071904, A129131, A129145.

from sympy import primepi, prime
def A129146(n):
    if n == 1: return -6
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return prime(n+1)-m # Chai Wah Wu, Aug 01 2024

a(n) = A065091(n) - A071904(n).

A129145 Alternately odd prime and odd composite numbers.

Original entry on oeis.org

3, 9, 5, 15, 7, 21, 11, 25, 13, 27, 17, 33, 19, 35, 23, 39, 29, 45, 31, 49, 37, 51, 41, 55, 43, 57, 47, 63, 53, 65, 59, 69, 61, 75, 67, 77, 71, 81, 73, 85, 79, 87, 83, 91, 89, 93, 97, 95, 101, 99, 103, 105, 107, 111, 109, 115, 113, 117, 127, 119, 131, 121, 137, 123, 139, 125

Offset: 1

Zak Seidov, Apr 01 2007

nonn

For small n's a(2n) >a(2n-1) (that is for small n's, n-th odd prime less than n-th odd composite number), while for large n's a(2n) A129146

Cf. A065091, A071904, A129131, A129146.

Module[{nn=100,pr,cm,len},pr=Prime[Range[2,nn+1]];cm=Select[Range[ 9,2nn+1,2],CompositeQ];len=Min[Length[pr],Length[cm]];Riffle[Take[ pr,len],Take[cm,len]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 10 2019 *)

from sympy import prime, primepi
def A129145(n):
    if n&1: return prime((n>>1)+2)
    if n==2: return 9
    r = n>>1
    m, k = r, primepi(r) + r + (r>>1)
    while m != k:
        m, k = k, primepi(k) + r + (k>>1)
    return m # Chai Wah Wu, Aug 01 2024

a(2n-1)=A065091(n), a(2n) =A071904(n).

cp's OEIS Frontend

A129146 a(n) = n-th odd prime minus n-th odd composite number.

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