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%I #32 Aug 27 2020 17:08:57
%S 1,2,3,4,5,3,7,8,9,10,11,6,13,14,5,16,17,9,19,20,21,22,23,12,25,26,27,
%T 28,29,5,31,32,33,34,7,9,37,38,39,40,41,21,43,44,15,46,47,24,49,50,51,
%U 52,53,27,55,56,57,58,59,10,61,62,63,64,65,33,67,68,69
%N Primorial deflation of n (numerator): 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 numerator of g(n).
%C See A319627 for the corresponding denominators.
%C The restriction of f to the natural numbers corresponds to A108951.
%C The function g is completely multiplicative over the positive rational numbers with g(2) = 2 and g(q) = q/p for any pair (p, q) of consecutive prime numbers.
%C The ratio A319626(n)/A319627(n) can be viewed as a "primorial deflation" of n (see also A329900), with the inverse operation being n = A108951(A319626(n)) / A108951(A319627(n)), where A319627(k) = 1 for all k in A025487. - _Daniel Suteu_, Dec 29 2019
%H Daniel Suteu, <a href="/A319626/b319626.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = n / gcd(n, A064989(n)) = n / A330749(n).
%F a(n) <= n with equality iff n belongs to A319630.
%F A006530(a(n)) = A006530(n).
%F A053585(a(n)) = A053585(n).
%F From _Antti Karttunen_, Dec 29 2019: (Start)
%F a(A108951(n)) = n.
%F a(A025487(n)) = A329900(A025487(n)) = A181815(n).
%F Many of the formulas given in A329900 apply here as well:
%F a(n!) = A307035(n), a(A002182(n)) = A329902(n), and so on.
%F (End)
%e f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 21.
%t Array[#1/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)); numerator(prod(i=1, #f~, my (p=f[i,1]); (p/if (p>2, precprime(p-1), 1))^f[i,2]))
%Y A left inverse of A108951. Coincides with A329900 on A025487.
%Y Cf. A006530, A053585, A064989, A181815, A307035, A319627, A319630, A329902, A330749, A330750 (rgs-transform), A330751 (ordinal transform).
%K nonn,frac
%O 1,2
%A _Rémy Sigrist_, Sep 25 2018
%E "Primorial deflation" prefixed to the name by _Antti Karttunen_, Dec 29 2019
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