A319626 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).

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, 28, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 69

Offset: 1

Author

Rémy Sigrist, Sep 25 2018

Keywords

• nonn
• frac

Comments

See A319627 for the corresponding denominators.

The restriction of f to the natural numbers corresponds to A108951.

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.

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

Examples

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

Links

• Daniel Suteu, Table of n, a(n) for n = 1..10000

Crossrefs

A left inverse of A108951. Coincides with A329900 on A025487.

Cf. A006530, A053585, A064989, A181815, A307035, A319627, A319630, A329902, A330749, A330750 (rgs-transform), A330751 (ordinal transform).

Programs

Mathematica

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

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]))

Formula

a(n) = n / gcd(n, A064989(n)) = n / A330749(n).

a(n) <= n with equality iff n belongs to A319630.

A006530(a(n)) = A006530(n).

A053585(a(n)) = A053585(n).

From Antti Karttunen, Dec 29 2019: (Start)

a(A108951(n)) = n.

a(A025487(n)) = A329900(A025487(n)) = A181815(n).

Many of the formulas given in A329900 apply here as well:

a(n!) = A307035(n), a(A002182(n)) = A329902(n), and so on.

(End)

Extensions

"Primorial deflation" prefixed to the name by Antti Karttunen, Dec 29 2019