

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.


105



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
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OFFSET

1,2


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 [reexpressed by Peter Munn, May 23 2020]
Not multiplicative in usual sense  but letting m=2n1=product_j (p_j)^(e_j) then a(n)=a((m+1)/2)=product_j (p_(j1))^(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)


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


FORMULA

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


EXAMPLE

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


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 *)


PROG

(Scheme) (define (A064216 n) (A064989 ( (+ n n) 1))) ;; Antti Karttunen, May 12 2014
(PARI) a(n) = {my(f = factor(2*n1)); 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


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 base2 representation of a(n), respectively).
Here obviously the variant 2, A151799(n) = A007917(n1), of the prevprime function is used.
Cf. also A003961, A270430, A270431.
Cf. also permutations A122111, A156552, A241909, A243071, A243065, A243505, A244153, A245611, A254116.
Sequence in context: A250472 A291588 A064620 * A075300 A329821 A259153
Adjacent sequences: A064213 A064214 A064215 * A064217 A064218 A064219


KEYWORD

easy,nonn


AUTHOR

Howard A. Landman, Sep 21 2001


EXTENSIONS

More terms from Reinhard Zumkeller, Sep 26 2001
Additional description added by Antti Karttunen, May 12 2014


STATUS

approved



