A253046 -id:A253046 - cp's OEIS frontend

A100484 The primes doubled; Even semiprimes.

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Reinhard Zumkeller, Nov 22 2004

Essentially the same as A001747.
Right edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
A253046(a(n)) > a(n). - Reinhard Zumkeller, Dec 26 2014
Apart from first term, these are the tau2-primes as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019
For every positive integer b and each m in this sequence b^(m-1) == b (mod m). - Florian Baur, Nov 26 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Semiprime

Subsequence of A091376. After the initial 4 also a subsequence of A039956.
Cf. A046315, A152099, A179740.
Cf. A001748, A253046, A353478 (characteristic function).
Row 3 of A286625, column 3 of A286623.

2*Filtered([1..300],IsPrime); # Muniru A Asiru, Oct 05 2018

List([1..70], n-> 2*Primes[n]); # G. C. Greubel, May 18 2019

a100484 n = a100484_list !! (n-1)
a100484_list = map (* 2) a000040_list
-- Reinhard Zumkeller, Jan 31 2012

[2*p: p in PrimesUpTo(350)]; // Vincenzo Librandi, Mar 27 2014

A100484:=n->2*ithprime(n); seq(A100484(n), n=1..70); # Wesley Ivan Hurt, Mar 27 2014

2*Prime[Range[70]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

2*primes(70) \\ Charles R Greathouse IV, Aug 21 2011

[2*nth_prime(n) for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = 2 * A000040(n).
a(n) = A001747(n+1).
n>1: A000005(a(n)) = 4; A000203(a(n)) = 3*A008864(n); A000010(a(n)) = A006093(n); intersection of A001358 and A005843.
a(n) = A116366(n-1, n-1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = A077017(n+1), n>1. - R. J. Mathar, Sep 02 2008
A078834(a(n)) = A000040(n). - Reinhard Zumkeller, Sep 19 2011
a(n) = A087112(n, 1). - Reinhard Zumkeller, Nov 25 2012
A000203(a(n)) = 3*n/2 + 3, n > 1. - Wesley Ivan Hurt, Sep 07 2013

Simpler definition.

A001748 a(n) = 3 * prime(n).

Original entry on oeis.org

6, 9, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753

Offset: 1

N. J. A. Sloane

Semiprimes divisible by 3. - Jianing Song, Oct 02 2022

Paolo P. Lava, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Skeleton

Subsequence of A001358.

Cf. A000040, A100484, A253046, A164023, A164024, A179545, A253046.

a001748 = (* 3) . a000040  -- Reinhard Zumkeller, Dec 26 2014

[3*p: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 26 2014

Prime[Range[22]]*3 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

3*primes(22) \\ Charles R Greathouse IV, May 19 2011

A164023(a(n)) = A164024(a(n)) = A000040(n). - Reinhard Zumkeller, Aug 09 2009
a(n) = 3*A000040(n). - Omar E. Pol, Jan 31 2012
A253046(a(n)) < a(n). - Reinhard Zumkeller, Dec 26 2014

A064614 Exchange 2 and 3 in the prime factorization of n.

Original entry on oeis.org

1, 3, 2, 9, 5, 6, 7, 27, 4, 15, 11, 18, 13, 21, 10, 81, 17, 12, 19, 45, 14, 33, 23, 54, 25, 39, 8, 63, 29, 30, 31, 243, 22, 51, 35, 36, 37, 57, 26, 135, 41, 42, 43, 99, 20, 69, 47, 162, 49, 75, 34, 117, 53, 24, 55, 189, 38, 87, 59, 90, 61, 93, 28, 729, 65, 66, 67, 153, 46

Offset: 1

Reinhard Zumkeller, Sep 25 2001

A self-inverse permutation of the natural numbers.
a(1) = 1, a(2) = 3, a(3) = 2, a(p) = p for primes p > 3 and a(u * v) = a(u) * a(v) for u, v > 0.
A permutation of the natural numbers: a(a(n)) = n for all n and a(n) = n iff n = 6^k * m for k >= 0 and m > 0 with gcd(m, 6) = 1 (see A064615).
A000244 and A000079 give record values and where they occur. - Reinhard Zumkeller, Feb 08 2010

			a(15) = a(3*5) = a(3)*a(5) = 2*5 = 10;
a(16) = a(2^4) = a(2)^4 = 3^4 = 81;
a(17) = 17;
a(18) = a(2*3^2) = a(2)*a(3^2) = 3*a(3)^2 = 3*2^2 = 12.

T. D. Noe, Table of n, a(n) for n = 1..1000
A. B. Frizell, Certain non-enumerable sets of infinite permutations, Bull. Amer. Math. Soc. 21, No. 10 (1915), 495-499.
Index entries for sequences that are permutations of the natural numbers.
Index to divisibility sequences.

Cf. A000079, A000244, A007814, A007949, A027746, A064615, A065330, A253046, A253047.

a064614 1 = 1
a064614 n = product $ map f $ a027746_row n where
   f 2 = 3; f 3 = 2; f p = p
-- Reinhard Zumkeller, Apr 09 2012, Jan 03 2011

a[n_] := Times @@ Power @@@ (FactorInteger[n] /. {2, e2_} -> {0, e2} /. {3, e3_} -> {2, e3} /. {0, e2_} -> {3, e2}); Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Nov 20 2012 *)
a[n_] := n * Times @@ ({3/2, 2/3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n)=my(x=valuation(n, 2)-valuation(n, 3)); n*2^-x*3^x \\ Charles R Greathouse IV, Jun 28 2015

from operator import mul
from functools import reduce
from sympy import factorint
def A064614(n):
    return reduce(mul,((5-p if 2<=p<=3 else p)**e for p,e in factorint(n).items())) if n > 1 else n
# Chai Wah Wu, Dec 27 2014

a(n) = A065330(n) * (2 ^ A007949(n)) * (3 ^ A007814(n)). - Reinhard Zumkeller, Jan 03 2011
Completely multiplicative with a(2) = 3, a(3) = 2, and a(p) = p for primes p > 3. - Charles R Greathouse IV, Jun 28 2015
Sum_{k=1..n} a(k) ~ (6/7) * n^2. - Amiram Eldar, Oct 28 2022
Dirichlet g.f.: zeta(s-1)*((2^s-2)*(3^s-3))/((2^s-3)*(3^s-2)). - Amiram Eldar, Dec 30 2022

A251561 A permutation of the natural numbers: interchange p and 2p for every prime p.

Original entry on oeis.org

1, 4, 6, 2, 10, 3, 14, 8, 9, 5, 22, 12, 26, 7, 15, 16, 34, 18, 38, 20, 21, 11, 46, 24, 25, 13, 27, 28, 58, 30, 62, 32, 33, 17, 35, 36, 74, 19, 39, 40, 82, 42, 86, 44, 45, 23, 94, 48, 49, 50, 51, 52, 106, 54, 55, 56, 57, 29, 118, 60, 122, 31, 63, 64, 65, 66

Offset: 1

N. J. A. Sloane, Dec 26 2014

nonn

a(A001751(n)) != A001751(n). - Reinhard Zumkeller, Dec 27 2014

Michael De Vlieger, 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, A253046.

Cf. A020639, A010051, A001751.

a251561 1 = 1
a251561 n | q == 1                    = 2 * p
          | p == 2 && a010051' q == 1 = q
          | otherwise                 = n
          where q = div n p; p = a020639 n
-- Reinhard Zumkeller, Dec 27 2014

a251561[n_] := Block[{f}, f[x_] := Which[PrimeQ[x], 2 x, PrimeQ[x/2], x/2, True, x]; Array[f, n]]; a251561[66] (* Michael De Vlieger, Dec 26 2014 *)

from sympy import isprime
def A251561(n):
    if n == 2:
        return 4
    q,r = divmod(n,2)
    if r :
        if isprime(n):
            return 2*n
        return n
    if isprime(q):
        return q
    return n # Chai Wah Wu, Dec 26 2014

A253106 Semiprimes with smallest factor <= 3.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 146, 158, 159, 166, 177, 178, 183, 194, 201, 202, 206, 213, 214, 218, 219, 226, 237, 249, 254, 262, 267, 274, 278

Offset: 1

Reinhard Zumkeller, Dec 26 2014

nonn

A253046(a(n)) != a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. Union of A001748 and A100484, A020639, A010051, A001358, A171156, A253046.

a253106 n = a253106_list !! (n-1)
a253106_list = filter f [1..] where
   f x = p <= 3 && a010051' (div x p) == 1  where p = a020639 x

A253047 Start with the natural numbers 1,2,3,...; interchange 2prime(i) and 3prime(i+1) for each i, and interchange prime(prime(i)) with prime(2*prime(i)) for each i.

Original entry on oeis.org

1, 2, 7, 9, 13, 15, 3, 8, 4, 21, 29, 12, 5, 33, 6, 16, 43, 18, 19, 20, 10, 39, 23, 24, 25, 51, 27, 28, 11, 30, 79, 32, 14, 57, 35, 36, 37, 69, 22, 40, 101, 42, 17, 44, 45, 87, 47, 48, 49, 50, 26, 52, 53, 54, 55, 56, 34, 93, 139, 60, 61, 111, 63

Offset: 1

N. J. A. Sloane, Dec 26 2014

nonn

This is an involution on the natural numbers: applying it twice gives the identity permutation.

Chai Wah Wu, 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, A253046, A253046.

f:= proc(t)
local r;
  if t mod 2 = 0 and isprime(t/2) then  3*nextprime(t/2)
  elif t mod 3 = 0 and isprime(t/3) then 2*prevprime(t/3)
  elif isprime(t) then
    r:= numtheory:-pi(t);
    if isprime(r) then ithprime(2*r)
    elif r mod 2 = 0 and isprime(r/2) then ithprime(r/2)
    else t
    fi
  else t
  fi
end proc:
seq(f(i),i=1..100); # Robert Israel, Dec 26 2014

f[t_] := Module[{r}, Which[EvenQ[t] && PrimeQ[t/2], 3 NextPrime[t/2], Divisible[t, 3] && PrimeQ[t/3], 2 NextPrime[t/3, -1], PrimeQ[t], r = PrimePi[t]; Which[PrimeQ[r], Prime[2r], EvenQ[r] && PrimeQ[r/2], Prime[r/2], True, t], True, t]];
Array[f, 100] (* Jean-François Alcover, Jul 27 2020, after Robert Israel *)

from sympy import isprime, primepi, prevprime, nextprime, prime
def A253047(n):
    if n <= 2:
        return n
    if n == 3:
        return 7
    q2, r2 = divmod(n,2)
    if r2:
        q3, r3 = divmod(n,3)
        if r3:
            if isprime(n):
                m = primepi(n)
                if isprime(m):
                    return prime(2*m)
                x, y = divmod(m,2)
                if not y:
                    if isprime(x):
                        return prime(x)
            return n
        if isprime(q3):
            return 2*prevprime(q3)
        return n
    if isprime(q2):
        return 3*nextprime(q2)
    return n # Chai Wah Wu, Dec 27 2014

Definition supplied by Robert Israel, Dec 26 2014

Offset changed to 1 by Chai Wah Wu, Dec 27 2014

cp's OEIS Frontend

A100484 The primes doubled; Even semiprimes.

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A001748 a(n) = 3 * prime(n).

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A064614 Exchange 2 and 3 in the prime factorization of n.

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A251561 A permutation of the natural numbers: interchange p and 2p for every prime p.

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A253106 Semiprimes with smallest factor <= 3.

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A253047 Start with the natural numbers 1,2,3,...; interchange 2*prime(i) and 3*prime(i+1) for each i, and interchange prime(prime(i)) with prime(2*prime(i)) for each i.

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Maple

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Extensions

A253047 Start with the natural numbers 1,2,3,...; interchange 2prime(i) and 3prime(i+1) for each i, and interchange prime(prime(i)) with prime(2*prime(i)) for each i.