A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 6, 1, 2, 1, 16, 1, 3, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 5, 1, 32, 1, 2, 1, 9, 1, 2, 1, 8, 1, 3, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 10, 1, 2, 1, 64, 1, 3, 1, 4, 1, 2, 1, 18, 1, 2, 1, 4, 1, 3, 1, 16, 1, 2, 1, 6, 1, 2, 1, 8, 1, 5, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 3, 1, 8, 1

Offset

1,2

Comments

When applied to arbitrary n, the "primorial deflation" (term coined by Matthew Vandermast in A181815) induces the splitting of n to two factors A328478(n)*A328479(n) = n, where we call A328478(n) the non-deflatable component of n (which is essentially discarded), while A328479(n) is the deflatable component. Only if n is in A025487, then the entire n is deflatable, i.e., A328478(n) = 1 and A328479(n) = n.

According to Daniel Suteu, also the ratio (A319626(n) / A319627(n)) can be viewed as a "primorial deflation". That definition coincides with this one when restricted to terms of A025487, as for all k in A025487, A319626(k) = a(k), and A319627(k) = 1. - Antti Karttunen, Dec 29 2019

Links

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

Formula

For odd n, a(n) = 1, for even n, a(n) = A000040(A276084(n)) * a(A111701(n)).

For even n, a(n) = A000040(A276084(n)) * a(n/A002110(A276084(n))).

A108951(a(n)) = A328479(n), for n >= 1.

a(A108951(n)) = n, for n >= 1.

a(A328479(n)) = a(n), for n >= 1.

a(A328478(n)) = 1, for n >= 1.

a(A002110(n)) = A000040(n), for n >= 1.

a(A000142(n)) = A307035(n), for n >= 0.

a(A283477(n)) = A019565(n), for n >= 0.

a(A329886(n)) = A005940(1+n), for n >= 0.

a(A329887(n)) = A163511(n), for n >= 0.

a(A329602(n)) = A329888(n), for n >= 1.

a(A025487(n)) = A181815(n), for n >= 1.

a(A124859(n)) = A181819(n), for n >= 1.

a(A181817(n)) = A025487(n), for n >= 1.

a(A181821(n)) = A122111(n), for n >= 1.

a(A002182(n)) = A329902(n), for n >= 1.

a(A260633(n)) = A329889(n), for n >= 1.

a(A033833(n)) = A330685(n), for n >= 1.

a(A307866(1+n)) = A330686(n), for n >= 1.

a(A330687(n)) = A330689(n), for n >= 1.

Mathematica

Array[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, 105] (* Michael De Vlieger, Dec 28 2019 *)
Array[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, 105] (* Michael De Vlieger, Jan 11 2020 *)

Prog

(PARI) A329900(n) = { my(m=1, pp=1); while(1, forprime(p=2, , if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); (m); };
(PARI)
A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276084(n) = { for(i=1, oo, if(n%prime(i), return(i-1))); }
A329900(n) = if(n%2, 1, prime(A276084(n))*A329900(A111701(n)));

Crossrefs

A left inverse of A108951. Coincides with A319626 on A025487.

Cf. A002110, A002182, A111701, A181815, A181817, A181819, A181821, A276084, A304886, A319626, A319627, A328478, A328479, A329889, A329902, A330685, A330686, A330689.

Sequence in context: A329644 A256578 A127461 * A101261 A214057 A067614

Adjacent sequences: A329897 A329898 A329899 * A329901 A329902 A329903

Keyword

nonn

Author

Antti Karttunen, Dec 22 2019

Status

approved