A251561 -id:A251561 - cp's OEIS frontend

A001751 Primes together with primes multiplied by 2.

2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163, 166

Offset: 1

N. J. A. Sloane

For n > 1, a(n) is position of primes in A026741.
For n > 1, a(n) is the position of the ones in A046079. - Ant King, Jan 29 2011
A251561(a(n)) != a(n). - Reinhard Zumkeller, Dec 27 2014
Number of terms <= n is pi(n) + pi(n/2). - Robert G. Wilson v, Aug 04 2017
Number of terms <=10^k: 7, 40, 263, 1898, 14725, 120036, 1013092, 8762589, 77203401, 690006734, 6237709391, 56916048160, 523357198488, 4843865515369, ..., . - Robert G. Wilson v, Aug 04 2017
Complement of A264828. - Chai Wah Wu, Oct 17 2024

T. D. Noe, Table of n, a(n) for n = 1..10000

Union of A001747 and A000040.
Subsequence of A039698 and of A033948.
Cf. A026741, A046079, A178156, A251561, A264828.

a001751 n = a001751_list !! (n-1)
a001751_list = 2 : filter (\n -> (a010051 $ div n $ gcd 2 n) == 1) [1..]
-- Reinhard Zumkeller, Jun 20 2011 (corrected, improved), Dec 17 2010

Select[Range[163], Or[PrimeQ[#], PrimeQ[1/2 #]] &] (* Ant King, Jan 29 2011 *)
upto=200;With[{pr=Prime[Range[PrimePi[upto]]]},Select[Sort[Join[pr,2pr]],# <= upto&]] (* Harvey P. Dale, Sep 23 2014 *)

isA001751(n)=isprime(n/gcd(n,2)) || n==2

list(lim)=vecsort(concat(primes(primepi(lim)), 2* primes(primepi(lim\2)))) \\ Charles R Greathouse IV, Oct 31 2012

from sympy import primepi
def A001751(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-primepi(x)-primepi(x>>1))
    return bisection(f,n,n) # Chai Wah Wu, Oct 17 2024

A253046 An involution of the natural numbers: if n = 2p_i then replace n with 3p_{i+1}, and conversely if n = 3p_i then replace n with 2p_{i-1}, where p_i denotes the i-th prime.

Original entry on oeis.org

1, 2, 3, 9, 5, 15, 7, 8, 4, 21, 11, 12, 13, 33, 6, 16, 17, 18, 19, 20, 10, 39, 23, 24, 25, 51, 27, 28, 29, 30, 31, 32, 14, 57, 35, 36, 37, 69, 22, 40, 41, 42, 43, 44, 45, 87, 47, 48, 49, 50, 26, 52, 53, 54, 55, 56, 34, 93, 59, 60, 61, 111, 63, 64, 65, 66, 67

Offset: 1

N. J. A. Sloane, Dec 26 2014

nonn

a(m) != m iff m is a term of A253106, i.e., a semiprime divisible by 2 or 3; a(A100484(n)) > A100484(n); a(A001748(n)) < A001748(n). - Reinhard Zumkeller, Dec 26 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. B. Frizell, Certain non-enumerable sets of infinite permutations. Bull. Amer. Math. Soc. 21 (1915), no. 10, 495-499.
Index entries for sequences that are permutations of the natural numbers

Cf. A064614, A251561, A020639, A049084, A000040, A253106, A001748, A100484.

a253046 n | i == 0 || p > 3 = n
          | p == 2          = 3 * a000040 (i + 1)
          | otherwise       = 2 * a000040 (i - 1)
            where i = a049084 (div n p);  p = a020639 n
-- Reinhard Zumkeller, Dec 26 2014

a253046[n_] := Block[{f},
  f[x_] := Which[PrimeQ[x/2], 3 NextPrime[x/2],
    PrimeQ[x/3], 2 NextPrime[x/3, -1],
True, x];Array[f, n]]; a253046[67] (* Michael De Vlieger, Dec 27 2014 *)

from sympy import isprime, nextprime, prevprime
def A253046(n):
    q2, r2 = divmod(n,2)
    if not r2 and isprime(q2):
        return 3*nextprime(q2)
    else:
        q3, r3 = divmod(n,3)
        if not r3 and isprime(q3):
            return 2*prevprime(q3)
        return n # Chai Wah Wu, Dec 27 2014

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A001751 Primes together with primes multiplied by 2.

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A253046 An involution of the natural numbers: if n = 2p_i then replace n with 3p_{i+1}, and conversely if n = 3p_i then replace n with 2p_{i-1}, where p_i denotes the i-th prime.

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cp's OEIS Frontend

A001751 Primes together with primes multiplied by 2.

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Author

Keywords

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Programs

Haskell

Mathematica

PARI

PARI

Python

A253046 An involution of the natural numbers: if n = 2*p_i then replace n with 3*p_{i+1}, and conversely if n = 3*p_i then replace n with 2*p_{i-1}, where p_i denotes the i-th prime.

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Programs

Haskell

Mathematica

Python

A253046 An involution of the natural numbers: if n = 2p_i then replace n with 3p_{i+1}, and conversely if n = 3p_i then replace n with 2p_{i-1}, where p_i denotes the i-th prime.