A319627 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).

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

Offset: 1

Author

Rémy Sigrist, Sep 25 2018

Keywords

• nonn
• look
• frac

Comments

See A319626 for the corresponding numerators and additional comments.

Examples

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

Links

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

Crossrefs

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

Cf. A319626 (numerators, see comments there).

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

Programs

Mathematica

Array[#2/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)); denominator(prod(i=1, #f~, my (p=f[i,1]); (p/if (p>2, precprime(p-1), 1))^f[i,2]))

Formula

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

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

Extensions

"Primorial deflation" prefixed to the name by Antti Karttunen, Apr 29 2022