A064216 Replace each p^e with prevprime(p)^e in the prime factorization of odd numbers; inverse of sequence A048673 considered as a permutation of the natural numbers.

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 10, 19, 9, 8, 23, 29, 14, 15, 31, 22, 37, 41, 12, 43, 25, 26, 47, 21, 34, 53, 59, 20, 33, 61, 38, 67, 71, 18, 35, 73, 16, 79, 39, 46, 83, 55, 58, 51, 89, 28, 97, 101, 30, 103, 107, 62, 109, 57, 44, 65, 49, 74, 27, 113, 82, 127, 85, 24, 131

Offset: 1

Author

Howard A. Landman, Sep 21 2001

Keywords

• easy
• nonn

Comments

a((A003961(n) + 1) / 2) = n and A003961(a(n)) = 2*n - 1 for all n. If the sequence is indexed by odd numbers only, it becomes multiplicative. In this variant sequence, denoted b, even indices don't exist, and we get b(1) = a(1) = 1, b(3) = a(2) = 2, b(5) = 3, b(7) = 5, b(9) = 4 = b(3) * b(3), ... , b(15) = 6 = b(3) * b(5), and so on. This property can also be stated as: a(x) * a(y) = a(((2x - 1) * (2y - 1) + 1) / 2) for x, y > 0. - Reinhard Zumkeller [re-expressed by Peter Munn, May 23 2020]

Not multiplicative in usual sense - but letting m=2n-1=product_j (p_j)^(e_j) then a(n)=a((m+1)/2)=product_j (p_(j-1))^(e_j). - Henry Bottomley, Apr 15 2005

From Antti Karttunen, Jul 25 2016: (Start)

Several permutations that use prime shift operation A064989 in their definition yield a permutation obtained from their odd bisection when composed with this permutation from the right. For example, we have:

A243505(n) = A122111(a(n)).

A243065(n) = A241909(a(n)).

A244153(n) = A156552(a(n)).

A245611(n) = A243071(a(n)).

(End)

Examples

			For n=11, the 11th odd number is 2*11 - 1 = 21 = 3^1 * 7^1. Replacing the primes 3 and 7 with the previous primes 2 and 5 gives 2^1 * 5^1 = 10, so a(11) = 10. - _Michael B. Porter_, Jul 25 2016
		

Links

• Carl R. White, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization.
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Odd bisection of A064989 and A252463.

Row 1 of A251721, Row 2 of A249821.

Cf. A048673 (inverse permutation), A048674 (fixed points).

Cf. A246361 (numbers n such that a(n) <= n.)

Cf. A246362 (numbers n such that a(n) > n.)

Cf. A246371 (numbers n such that a(n) < n.)

Cf. A246372 (numbers n such that a(n) >= n.)

Cf. A246373 (primes p such that a(p) >= p.)

Cf. A246374 (primes p such that a(p) < p.)

Cf. A246343 (iterates starting from n=12.)

Cf. A246345 (iterates starting from n=16.)

Cf. A245448 (this permutation "squared", a(a(n)).)

Cf. A253894, A254044, A254045 (binary width, weight and the number of nonleading zeros in base-2 representation of a(n), respectively).

Cf. A285702, A285703 (phi and sigma applied to a(n).)

Here obviously the variant 2, A151799(n) = A007917(n-1), of the prevprime function is used.

Cf. also A003961, A270430, A270431.

Cf. also permutations A122111, A156552, A241909, A243071, A243065, A243505, A244153, A245611, A254116.

Programs

Mathematica

Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 69}] (* Michael De Vlieger, Dec 18 2014, revised Mar 17 2016 *)

PARI

a(n) = {my(f = factor(2*n-1)); for (k=1, #f~, f[k,1] = precprime(f[k,1]-1)); factorback(f);} \\ Michel Marcus, Mar 17 2016

Python

from sympy import factorint, prevprime
from operator import mul
def a(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f]) # Indranil Ghosh, May 13 2017

Scheme

(define (A064216 n) (A064989 (- (+ n n) 1))) ;; Antti Karttunen, May 12 2014

Formula

a(n) = A064989(2n - 1). - Antti Karttunen, May 12 2014

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime > 2} ((p^2-p)/(p^2-q(p))) = 0.6621117868..., where q(p) = prevprime(p) (A151799). - Amiram Eldar, Jan 21 2023

Extensions

More terms from Reinhard Zumkeller, Sep 26 2001

Additional description added by Antti Karttunen, May 12 2014