A307866 -id:A307866 - cp's OEIS frontend

A330686 Primorial deflation of (nonzero) K-champion numbers: a(n) is the unique integer x such that A108951(x) = A307866(1+n).

1, 4, 3, 8, 6, 12, 9, 24, 18, 48, 20, 36, 96, 40, 72, 30, 54, 80, 144, 60, 108, 160, 288, 120, 216, 320, 90, 576, 240, 432, 180, 480, 864, 360, 960, 720, 1920, 540, 252, 1440, 3840, 1080, 504, 2880, 1200, 2160, 1008, 5760, 2688, 2400, 4320, 2016, 11520, 3240, 4800, 1512, 8640, 4032, 23040, 1680, 6480, 9600, 3024, 17280, 8064, 7200

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..761 (computed from the b-file of A307866)

Cf. A074206, A181815, A307866, A329900, A353568.

a(n) = A329900(A307866(1+n)).

a(n) = A122111(A353568(n)). - Antti Karttunen, May 20 2022

A333963 Intersection of A307866 and A333953, together with the number 2.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 51840, 60480, 69120, 103680, 120960, 172800, 207360, 241920, 345600, 362880, 414720, 483840, 725760

Offset: 1

Amiram Eldar and David A. Corneth, Apr 15 2020

nonn

Numbers that are both recursively superabundant numbers and K-champion numbers.

Cf. A307866, A333952, A333953.

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.

Original entry on oeis.org

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.

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)).

A333952 Recursively highly composite numbers: numbers m such that A067824(m) > A067824(k) for all k < m.

Original entry on oeis.org

1, 2, 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, Apr 11 2020

nonn

This sequence is not to be confused with A333931.
The corresponding record values are 1, 2, 4, 6, 8, 16, 40, 52, 96, ...
Fink (2019) defined this sequence. He asked whether 720 is the largest term that is also highly composite number (A002182).
This is, except the terms 2, the sequence records of indices of A074206 for positive n as a(n) = 2*A074206(n), n>1, i.e. A307866. (formula from - Vladeta Jovovic, Jul 03 2005) - David A. Corneth, Apr 13 2020

			The first 6 terms of A067824 are 1, 2, 2, 4, 2, 6. The record values occur at 1, 2, 4, 6, the first 4 terms of this sequence.

David A. Corneth, Table of n, a(n) for n = 1..291 (first 120 terms from Amiram Eldar)
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See section 5.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A002182, A067824, A307866, A333953.

d[1] = 1; d[n_] := d[n] = 1 + DivisorSum[n, d[#] &, # < n &]; seq={}; dm = 0; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A307867 Numbers m such that K(m)/m > K(j)/j for all j < m, where K(m) is the Kalmár function (A074206).

Original entry on oeis.org

1, 72, 96, 144, 240, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 138240, 207360, 241920, 311040, 345600, 362880, 414720, 483840, 622080, 725760, 829440, 967680, 1244160, 1451520, 1935360, 2073600, 2419200

Offset: 1

Amiram Eldar, May 02 2019

nonn

Subsequence of A307866.

Amiram Eldar, Table of n, a(n) for n = 1..393

Cf. A004394, A163272, A307866.

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

cp's OEIS Frontend

A330686 Primorial deflation of (nonzero) K-champion numbers: a(n) is the unique integer x such that A108951(x) = A307866(1+n).

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A333963 Intersection of A307866 and A333953, together with the number 2.

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

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A333952 Recursively highly composite numbers: numbers m such that A067824(m) > A067824(k) for all k < m.

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A307867 Numbers m such that K(m)/m > K(j)/j for all j < m, where K(m) is the Kalmár function (A074206).

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