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%I #72 Jun 26 2023 10:16:43
%S 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,
%T 47,21,34,53,59,20,33,61,38,67,71,18,35,73,16,79,39,46,83,55,58,51,89,
%U 28,97,101,30,103,107,62,109,57,44,65,49,74,27,113,82,127,85,24,131
%N 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.
%C 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]
%C 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
%C From _Antti Karttunen_, Jul 25 2016: (Start)
%C 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:
%C A243505(n) = A122111(a(n)).
%C A243065(n) = A241909(a(n)).
%C A244153(n) = A156552(a(n)).
%C A245611(n) = A243071(a(n)).
%C (End)
%H Carl R. White, <a href="/A064216/b064216.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F a(n) = A064989(2n - 1). - _Antti Karttunen_, May 12 2014
%F 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
%e 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
%t 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 *)
%o (Scheme) (define (A064216 n) (A064989 (- (+ n n) 1))) ;; _Antti Karttunen_, May 12 2014
%o (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
%o (Python)
%o from sympy import factorint, prevprime
%o from operator import mul
%o def a(n):
%o f=factorint(2*n - 1)
%o return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f]) # _Indranil Ghosh_, May 13 2017
%Y Odd bisection of A064989 and A252463.
%Y Row 1 of A251721, Row 2 of A249821.
%Y Cf. A048673 (inverse permutation), A048674 (fixed points).
%Y Cf. A246361 (numbers n such that a(n) <= n.)
%Y Cf. A246362 (numbers n such that a(n) > n.)
%Y Cf. A246371 (numbers n such that a(n) < n.)
%Y Cf. A246372 (numbers n such that a(n) >= n.)
%Y Cf. A246373 (primes p such that a(p) >= p.)
%Y Cf. A246374 (primes p such that a(p) < p.)
%Y Cf. A246343 (iterates starting from n=12.)
%Y Cf. A246345 (iterates starting from n=16.)
%Y Cf. A245448 (this permutation "squared", a(a(n)).)
%Y Cf. A253894, A254044, A254045 (binary width, weight and the number of nonleading zeros in base-2 representation of a(n), respectively).
%Y Cf. A285702, A285703 (phi and sigma applied to a(n).)
%Y Here obviously the variant 2, A151799(n) = A007917(n-1), of the prevprime function is used.
%Y Cf. also A003961, A270430, A270431.
%Y Cf. also permutations A122111, A156552, A241909, A243071, A243065, A243505, A244153, A245611, A254116.
%K easy,nonn
%O 1,2
%A _Howard A. Landman_, Sep 21 2001
%E More terms from _Reinhard Zumkeller_, Sep 26 2001
%E Additional description added by _Antti Karttunen_, May 12 2014
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