A328479 -id:A328479 - cp's OEIS frontend

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

Antti Karttunen, Dec 22 2019

nonn

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

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

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.

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 *)

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); };

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)));

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.

A328478 Divide n by the largest primorial that divides it and repeat until a fixed point is reached; a(n) is the fixed point.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 1, 25, 13, 27, 7, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 1, 73, 37, 75, 19, 77, 13, 79, 5, 81, 41, 83, 7, 85, 43, 87, 11, 89

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial numbers

Cf. A007814 (gives the number of iterations to reach a fixed point), A025487 (indices of 1's).
Cf. A002110, A053589, A111701, A328479.
Cf. also A093411 for analogous sequence.

A111701[n_] := A111701[n] = Block[{m = n, k = 1}, While[IntegerQ[m/Prime[k]], m = m/Prime[k]; k++]; m];
a[n_] := a[n] = If[A111701[n] == n, n, a[A111701[n]]];
Array[a, 105] (* Jean-François Alcover, Jan 11 2022, after Robert G. Wilson v in A111701 *)

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))); };

If A111701(n) == n, then a(n) = n, otherwise a(n) = a(A111701(n)).

a(n) = n / A328479(n).

Definition clarified by N. J. A. Sloane, Jan 19 2021

A340346 The largest divisor of n that is a term of A055932 (numbers divisible by all primes smaller than their largest prime factor).

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 16, 1, 2, 1, 12

Offset: 1

Peter Munn, Jan 04 2021

			For n=2: the largest divisor of 2 is 2, and 2 qualifies as divisible by all primes smaller than its largest prime factor, 2 (since there are no smaller primes). So a(2) = 2.
For n=42: of 42's divisors, no multiples of 7 qualify as being divisible by all primes smaller than their largest prime factor (since that factor is 7 and no divisor of 42 is divisible by 5, a smaller prime). The largest of 42's other divisors is 6, which qualifies (since it is divisible by 2, the only prime smaller than 6's largest prime factor, 3). So a(42) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

A003961, A006519, A055932, A064989, A341629 are used in a definition of this sequence.
Sequences with related definitions: A327832, A328479.
Cf. A234959.

a[?OddQ] = 1; a[n] := Module[{f = FactorInteger[n]}, ind = Position[PrimePi /@ First /@ f - Range @ Length[f], ?(# > 0 &)]; If[ind == {}, n, Times @@ Power @@@ f[[1 ;; ind[[1, 1]] - 1]]]]; Array[a, 100] (* _Amiram Eldar, Jan 14 2021 *)

is(n) = my(f=factor(n)[, 1]~); f==primes(#f); \\ A055932
a(n) = vecmax(select(is, divisors(n))); \\ Michel Marcus, Jan 19 2021

A341629(n) = if(1==n,1,my(f=factor(n)[, 1]~); (primepi(f[#f])==#f));
A340346(n) = fordiv(n,d,if(A341629(n/d),return(n/d))); \\ Antti Karttunen, Feb 25 2021

For n >= 1, a(2n-1) = 1, a(2n) = A006519(2n) * A003961(a(A064989(2n))).

For n >= 1, lcm(A006519(n), A234959(n)) | a(n).

A329379 a(n) = n/A093411(n), where A093411(n) is obtained by repeatedly dividing n by the largest factorial that divides it until an odd number is reached.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 6, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 6, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 96, 1, 2, 1, 4, 1, 6, 1, 8, 1

Offset: 1

Antti Karttunen, Nov 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..40319

Cf. A000142, A076934, A093411.

Cf. also A328479.

A076934(n) = for(k=2, oo , if(n%k, return(n), n /= k));
A093411(n) = if(!n,n,my(u=A076934(n)); if(n%2, return(n), return(A093411(u))));
A329379(n) = (n/A093411(n));

a(n) = n/A093411(n).

cp's OEIS Frontend

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.

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A328478 Divide n by the largest primorial that divides it and repeat until a fixed point is reached; a(n) is the fixed point.

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A340346 The largest divisor of n that is a term of A055932 (numbers divisible by all primes smaller than their largest prime factor).

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A329379 a(n) = n/A093411(n), where A093411(n) is obtained by repeatedly dividing n by the largest factorial that divides it until an odd number is reached.

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