A330689 -id:A330689 - 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.

A050377 Number of ways to factor n into "Fermi-Dirac primes" (members of A050376).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 4, 1, 1, 1, 2, 1

Offset: 1

Christian G. Bower, Nov 15 1999

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..100000 (the first 10000 terms from Reinhard Zumkeller)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000123, A001055, A002110, A005117, A018819, A050376-A050380, A025487.

Cf. A108951, A330687 (positions of records), A330688 (record values), A330689, A330690, A382295.

A018819:= proc(n) option remember;
  if n::odd then procname(n-1)
  else procname(n-1) + procname(n/2)
  fi
end proc:
A018819(0):= 1:
f:= n -> mul(A018819(s[2]),s=ifactors(n)[2]):
seq(f(n),n=1..100); # Robert Israel, Jan 14 2016

b[0] = 1; b[n_] := b[n] = b[n - 1] + If[EvenQ[n], b[n/2], 0];
a[n_] := Times @@ (b /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 27 2018 *)

A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2])); \\ Antti Karttunen, Dec 28 2019

Dirichlet g.f.: Product_{n in A050376} (1/(1-1/n^s)).
a(p^k) = A000123([k/2]) for all primes p.
a(A002110(n)) = 1.
Multiplicative with a(p^e) = A018819(e). - Christian G. Bower and David W. Wilson, May 22 2005
a(n) = Sum{a(d): d^2 divides n}, a(1) = 1. - Reinhard Zumkeller, Jul 12 2007
a(A108951(n)) = A330690(n). - Antti Karttunen, Dec 28 2019
a(n) = 1 for all squarefree values of n (A005117). - Eric Fox, Feb 03 2020
G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Nov 25 2020
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.7876368001694456669... (A382295), where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)). - Amiram Eldar, Oct 03 2023

More terms from Antti Karttunen, Dec 28 2019

A330687 Positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

Original entry on oeis.org

1, 4, 16, 64, 144, 256, 576, 1024, 1296, 2304, 5184, 9216, 20736, 82944, 186624, 331776, 746496, 1327104, 2073600, 2985984, 5308416, 8294400, 18662400, 21233664, 26873856, 33177600, 47775744, 51840000, 74649600, 107495424, 132710400, 207360000, 429981696, 530841600, 671846400, 829440000, 1194393600, 1719926784, 1866240000, 2687385600

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

From David A. Corneth, Dec 29 2019: (Start)
Each term is a perfect square. Proof: A050377(n) is multiplicative with a(p^e) = A018819(e) and A018819(2k) = A018819(2k+1) and this sequence considers just records so we only need exponents of the form 2k; i.e., terms are squares.
Furthermore, the exponent 2 occurs at most once in the prime factorization of a(n) as A018819(2)^2 = A018819(4) = 4. So if the last two exponents in the prime factorization of m are 2's then setting the first of those two exponents to 4 and the other to 0 gives the same A050377(m).
Example of an application of this proof: we have 3600 = 2^4 * 3^2 * 5^2. We see the last two exponents are 2's so we can set the first of those two to 4 and the second to 0. This gives 2^4 * 3^4 = 1296 and, indeed, A050377(1296) = A050377(3600) = 16.
It seems that most exponents of a(n) are divisible by 4.
More specifically: Let S(n) be the list, possibly with duplicates, of exponents occurring in the prime factorizations of terms with the sum of exponents in the prime factorization <= n.
Let R(n) = |{x : x==4, S(n)}| / |S(n)|.
For example, S(8) is found from the following terms: 4, 16, 64, 144, 256, 576 and 1296 as the exponents in the prime factorization are (2), (4), (6), (4, 2), (8), (6, 2), (4, 4). The sums of each of these exponents per term is <= 8. There are 10 exponents listed. Of these 10 there are 5 that are divisible by 4. Therefore R(8) = 5/10.
Then it seems that R(n) tends to some value > 0.8 as n grows. (End)

David A. Corneth, Table of n, a(n) for n = 1..1004 (terms <= 10^100; first 294 terms from Antti Karttunen)

Cf. A018819, A050376, A050377, A329900, A330684 (square roots), A330688 (the record values), A330689 (primorial deflation).

Subsequence of A025487.

Block[{s = Rest@ Nest[Function[{a, n, b}, Append[a, {Times @@ Map[a[[# + 1, -1]] &, FactorInteger[n][[All, -1]] ], b}]] @@ {#1, #2, #1[[-1, -1]] + If[EvenQ@ #2, #1[[#2/2 + 1, -1]], 0 ]} & @@ {#, Length@ #} &, {{0, 1}, {1, 1}}, 10^5][[All, 1]], t}, t = Union@ FoldList[Max, s]; Map[FirstPosition[s, #][[1]] &, t]] (* Michael De Vlieger, Dec 29 2019 *)

upto_e = 101; \\ 101 --> 211 terms.
A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
v018819 = vector(upto_e,n,A018819(n)); \\ Precompute.
A050377(n) = factorback(apply(e -> v018819[e], factor(n)[, 2]));
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A330687list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, m=0, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,#v025487,if((t=A050377(v025487[i]))>m, listput(lista,v025487[i]); m=t)); Vec(lista); };
v330687 = A330687list(upto_e);
A330687(n) = v330687[n];
for(n=1,#v330687,print1(A330687(n),", "));

A050377(a(n)) = A330688(n).
A329900(a(n)) = A330689(n).
a(n) = A330684(n)^2.

A330684 Square roots of the positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 36, 48, 72, 96, 144, 288, 432, 576, 864, 1152, 1440, 1728, 2304, 2880, 4320, 4608, 5184, 5760, 6912, 7200, 8640, 10368, 11520, 14400, 20736, 23040, 25920, 28800, 34560, 41472, 43200, 51840, 57600, 82944, 86400, 100800, 103680, 115200, 129600, 172800, 207360, 230400, 259200, 345600, 388800, 403200

Offset: 1

Antti Karttunen, Dec 29 2019

nonn

Like terms in A330687, also these are all found in A025487.

Antti Karttunen, Table of n, a(n) for n = 1..1004 (computed from the b-file David A. Corneth computed for A330687)

Cf. A000196, A050376, A050377, A108951, A329900, A330687, A330689.

Subsequence of A025487.

PARI
```
A330684(n) = sqrtint(A330687(n));
```

a(n) = A000196(A330687(n)).
A108951(a(n)) = A000196(A330689(n)).
a(n) = A329900(A000196(A330689(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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A050377 Number of ways to factor n into "Fermi-Dirac primes" (members of A050376).

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A330687 Positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

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A330684 Square roots of the positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

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