A330751 -id:A330751

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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).

+10
36

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

LINKS

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

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)

EXAMPLE

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

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 *)

PROG

(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]))

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).

KEYWORD

nonn,frac

AUTHOR

Rémy Sigrist, Sep 25 2018

EXTENSIONS

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

STATUS

approved

A330752

Number of values of k, 1 <= k <= n, with A328478(k) = A328478(n), where A328478(n) gives the remainder when all maximal primorial divisors of n (from the largest to smallest) have been divided out.

+10
2

1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 1, 6, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 3, 1, 9, 1, 10, 1, 2, 1, 11, 1, 2, 1, 4, 1, 4, 1, 3, 1, 2, 1, 12, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 13, 1, 2, 1, 14, 1, 4, 1, 3, 1, 2, 1, 15, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 1, 6, 1, 2, 1, 5, 1, 3, 1, 3, 1, 2, 1, 16, 1, 2, 1, 3, 1, 4, 1, 5, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Ordinal transform of A328478.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

MATHEMATICA

A111701[n_] := A111701[n] = Block[{m = n, k = 1}, While[IntegerQ[m/Prime[k]], m = m/Prime[k]; k++]; m];

A328478[n_] := A328478[n] = If[A111701[n] == n, n, A328478[A111701[n]]];

Module[{b}, b[_] = 0;

a[n_] := With[{t = A328478[n]}, b[t] = b[t] + 1]];

Array[a, 105] (* Jean-François Alcover, Jan 11 2022, after Robert G. Wilson v in A111701 *)

PROG

(PARI)

up_to = 65537;

ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));

A328478(n) = { my(u=A111701(n)); if(u==n, return(n), return(A328478(u))); };

v330752 = ordinal_transform(vector(up_to, n, A328478(n)));

A330752(n) = v330752[n];

CROSSREFS

Cf. A111701, A328478, A330751.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 30 2019

STATUS

approved

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