A319627 - OEIS

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

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OFFSET

1,3

COMMENTS

See A319626 for the corresponding numerators and additional comments.

LINKS

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

FORMULA

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

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

EXAMPLE

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

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 *)

PROG

(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]))

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.

Sequence in context: A168008 A178810 A374237 * A334990 A217668 A119479

Adjacent sequences: A319624 A319625 A319626 * A319628 A319629 A319630

KEYWORD

nonn,look,frac

AUTHOR

Rémy Sigrist, Sep 25 2018

EXTENSIONS

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

STATUS

approved