A330686 -id:A330686 - 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.

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

Original entry on oeis.org

0, 1, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset: 1

Amiram Eldar, May 02 2019

nonn

The corresponding record values are 0, 1, 2, 3, 4, 8, 20, 26, 48, 76, 112, 132, 208, ... (see the link for more values).
Deléglise et al. (2008) proved that the number of powerful (A001694) terms in this sequence is finite. They ask whether a(391) = 485432135516160000 (the 112th powerful term) is the largest. - Amiram Eldar, Aug 20 2019
Is abs(omega(a(n)) - omega(a(n+1))) <= 1? (Cf. A001221.) - David A. Corneth, Apr 16 2020

David A. Corneth, Table of n, a(n) for n = 1..884 (terms 1..762 from Amiram Eldar, calculated by Deléglise et al.)
David A. Corneth, Table of n, a(n) for n = 1..1544 if abs(omega(a(n)) - omega(a(n + 1))) <= 1
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, Journal of Number Theory, Vol. 128, No. 6 (2008), pp. 1676-1716.
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Tables des 761 premiers champions de la fonction de Kalmar, alternative link.
Amiram Eldar, Table of n, a(n), A074206(a(n)), calculated by Deléglise et al.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A001221, A001694, A002093, A033833, A074206, A163272, A330686 (after primorial deflation).

a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; s = {}; am=-1; Do[a1 = a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 0, 10000}]; s

For n >= 1, a(1+n) = A108951(A330686(n)). - Antti Karttunen, Dec 31 2019

A353568 Prime shadows of (nonzero) K-champion numbers.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 9, 14, 15, 22, 20, 21, 26, 28, 33, 30, 35, 44, 39, 42, 55, 52, 51, 66, 65, 68, 70, 57, 78, 85, 110, 102, 95, 130, 114, 170, 138, 182, 220, 190, 174, 238, 260, 230, 255, 266, 340, 290, 276, 285, 322, 380, 310, 418, 345, 476, 406, 460, 370, 510, 506, 435, 532, 434, 580, 483, 570, 638, 465, 644, 518

Offset: 1

Antti Karttunen, Apr 29 2022

nonn

Sequence is injective (no duplicate values occur) because A307866 (after its initial zero) is a subsequence of A025487.

The finite number of powerful (A001694) terms in A307866 implies a finite number of odd terms in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..883

Cf. A025487, A122111, A181819, A307866, A330686.

Cf. also A353561, A353562.

v307866 = readvec("b307866_to.txt";) \\ Prepared from the b-file of A307866 with gawk ' { print $2 } ' < b307866.txt > b307866_to.txt
A307866(n) = v307866[n];
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A353568(n) = A181819(A307866(1+n));

a(n) = A181819(A307866(1+n)).

a(n) = A122111(A330686(n)).

A329889 a(n) is the unique integer k such that A108951(k) = A260633(n).

Original entry on oeis.org

1, 3, 6, 12, 5, 10, 20, 15, 30, 60, 28, 45, 21, 56, 90, 42, 180, 84, 63, 168, 70, 126, 140, 252, 189, 280, 504, 210, 378, 264, 1008, 420, 315, 220, 840, 630, 1680, 792, 330, 1260, 1584, 945, 1400, 660, 2520, 495, 1890, 882, 1320, 2100, 990, 1764, 2640, 4200, 1980, 3528, 1485, 2200, 8400, 2646, 3960, 6300, 2970, 5292, 7920, 3300, 5940, 2772

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2011 (computed from the b-file of A260633)

Cf. A008480, A050382, A108951, A260633, A329900.

Cf. also A329902, A330686.

a(n) = A329900(A260633(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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A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

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A353568 Prime shadows of (nonzero) K-champion numbers.

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A329889 a(n) is the unique integer k such that A108951(k) = A260633(n).

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