%I #24 Nov 04 2022 20:10:34

%S 1,1,2,1,3,1,5,1,4,3,7,1,11,5,2,1,13,2,17,3,10,7,19,1,9,11,8,5,23,1,

%T 29,1,14,13,3,1,31,17,22,3,37,5,41,7,4,19,43,1,25,9,26,11,47,4,21,5,

%U 34,23,53,1,59,29,20,1,33,7,61,13,38,3,67,1,71,31,6

%N Primorial deflation of n (denominator): Let f be the completely multiplicative function over the positive rational numbers defined by f(p) = A034386(p) for any prime number p; f constitutes a permutation of the positive rational numbers; let g be the inverse of f; for any n > 0, a(n) is the denominator of g(n).

%C See A319626 for the corresponding numerators and additional comments.

%H Daniel Suteu, <a href="/A319627/b319627.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A064989(n) / gcd(n, A064989(n)).

%F a(n) = 1 iff n belongs to A025487.

%e f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 5.

%t Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &, 120] (* _Michael De Vlieger_, Aug 27 2020 *)

%o (PARI) a(n) = my (f=factor(n)); denominator(prod(i=1, #f~, my (p=f[i,1]); (p/if (p>2, precprime(p-1), 1))^f[i,2]))

%Y Cf. A025487 (positions of 1's), A064989, A329900, A358217 [= bigomega(a(n))].

%Y Cf. A319626 (numerators, see comments there).

%Y Cf. also A307035, A337377, A348990 [= a(A003961(n))], A349169 (odd numbers k such that A348993(k) = a(k)), A354365/A354366.

%K nonn,look,frac

%O 1,3

%A _Rémy Sigrist_, Sep 25 2018

%E "Primorial deflation" prefixed to the name by _Antti Karttunen_, Apr 29 2022