Jimin Park - cp's OEIS frontend

A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

1, 1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16, 18, 20, 23, 25, 28, 30, 33, 35, 38, 41, 44, 47, 50, 53, 56, 60, 63, 67, 70, 74, 77, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 129, 134, 138, 143, 147, 152, 156, 161, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220

Offset: 1

Jimin Park, Nov 24 2024

nonn

For (n, n), there is at least one maximal partition P that's symmetric: (x, y) in P <=> (y, x) in P. This can be proven by manipulating integer sequences c(i) (i >= 1) such that 0 <= c(i) <= i+1 for all i and Sum_{i > 0} i*c(i) = 2n, which correspond to partitions P of (n, n) with size |P| = Sum_{i > 0} c(i), where c(i) is equal to number of (x, y) in P such that x+y = i.

			For n = 8, a(n) = 9, as (9, 9) can be expressed as the sum (0, 1) + (0, 2) + (0, 3) + (1, 0) + (2, 0) + (3, 0) + (1, 2) + (2, 1), but the longest sum for (8, 8) has 7 pairs.

Maximal size among partitions considered by A054242 and A201377.
Minimal x such that A378126(x, x) >= n.
Cf. A086435.

import math
def A378379(n: int) -> int:
  l = (math.isqrt(1+8*n)-1)//2 # l = A003056(n), min. possible largest pair norm
  r = n - (l-1)*(l+2)//2 # r = n - A000096(l-1), number of pairs with norm l
  return ((l-1)*l*(l+1)//3 + l*r + 1)//2 # ceil((A007290(l+1) + l*r) / 2)

a(n*(n+3)/2) = n*(n+1)*(n+2)/6.

A378126 Array read by antidiagonals: T(n, m) is the maximal size of partitions of (n, m) into sums of distinct pairs of nonnegative integers, excluding (0, 0).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 5, 4, 4, 4, 3, 3, 4, 5, 5, 5, 5, 5, 5, 4, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 5, 5, 5, 4, 4, 5, 5, 6, 6, 6, 6

Offset: 0

Jimin Park, Nov 17 2024

A378379(n) is the least number x such that T(x, x) >= n, and as the growth rate of A378379(n) is Theta(n^(3/2)), the growth rate of T(n, m) is O((n+m)^(2/3)).

			Table begins:
  0 1 1 2 2 2 3 3 3 3 ...
  1 2 2 3 3 3 4 4 4 4 ...
  1 2 3 3 4 4 4 5 5 5 ...
  2 3 3 4 4 4 5 5 5 6 ...
  2 3 4 4 5 5 5 6 6 6 ...
  2 3 4 4 5 5 6 6 6 7 ...
  3 4 4 5 5 6 6 6 7 7 ...
  3 4 5 5 6 6 6 7 7 7 ...
  3 4 5 5 6 6 7 7 7 8 ...
  3 4 5 6 6 7 7 7 8 8 ...
  ...
T(9, 5) = 7, as (9, 5) can be expressed as the sum (0, 1) + (0, 2) + (1, 0) + (1, 1) + (2, 0) + (2, 1) + (3, 0), which is the longest for (9, 5).

Jimin Park, Table of n, a(n) for n = 0..1000

Maximal size among partitions considered by A054242 and A201377.
The first row is T(n, 0) = A003056(n).
Cf. A086435, A378379.

import functools
@functools.cache
def A378126(n: int, m: int, t: tuple[int, int] = (0, 0)) -> int:
  if (n, m) <= t: return 0
  v = 1
  for nn in range(t[0], n//2+1):
    for nm in range(m+1):
      if (nn, nm) <= t: continue
      rn, rm = n-nn, m-nm
      if (rn, rm) <= (nn, nm): continue
      nv = 1 + A378126(rn, rm, (nn, nm))
      if nv > v: v = nv
  return v

T(n, m) = T(m, n).
T(n, m) >= T(n, 0) + T(0, m).
T(n, m) = A086435(p^n * q^m) for any distinct primes p and q.
T(n, 0) = A003056(n).
T(n, 1) = T(n, 0) + 1.
T(n, 2) = T(n-1, 0) + 2, for n >= 1.
T(n, m) = O((n+m)^(2/3)).

A193865 a(n) = (A187108(n+1)-A187108(n))/2.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 3

Offset: 1

Jimin Park, Aug 07 2011

nonn

var a:Array=new Array();
var i:int;
var n:int=0;
var ni:int;
var b:Array = new Array();
for (i=0; i<=1000; i++){
while(a[n]!=undefined) n++;
b.push(String(2*n+1));
a[n]=1;
ni=2*n+1;
while(ni>=2*n+1&&ni>1){
ni=3*ni+1;
while(ni%2==0)ni/=2;
a[(ni-1)/2]=1;
}}
for(i=0;i<1000;i++)b[i]=(b[i+1]-b[i])/2;b.pop();
trace(b);

A187108 Odd numbers not in the trajectory of a smaller number under the Collatz (3x+1) iteration.

Original entry on oeis.org

1, 3, 7, 9, 15, 19, 21, 25, 27, 33, 37, 39, 43, 45, 51, 55, 57, 63, 69, 73, 75, 79, 81, 87, 93, 97, 99, 105, 109, 111, 115, 117, 123, 127, 129, 133, 135, 141, 145, 147, 151, 153, 159, 163, 165, 169, 171, 177, 181, 183, 187, 189, 195, 199, 201, 207, 213, 217

Offset: 1

Jimin Park, Mar 05 2011

nonn

These are the odd numbers in A061641, which gives a more detailed explanation. - T. D. Noe, Mar 05 2011

Cf. A061641.

// AS3
var a:Array=new Array();
var i:int;
var n:int=0;
var ni:int;
var s:String='';
for (i=0;i<50;i++){
while(a[n]!=undefined) n++;
s+=String(2*n+1)+',';
a[n]=1;
ni=2*n+1;
while(ni>=2*n+1&&ni>1){
ni=3*ni+1;
while(ni%2==0)ni/=2;
a[(ni-1)/2]=1;
}
}
trace(s);

A166549 The number of halving steps of the Collatz 3x+1 map to reach 1 starting from 2n-1.

Original entry on oeis.org

0, 5, 4, 11, 13, 10, 7, 12, 9, 14, 6, 11, 16, 70, 13, 67, 18, 10, 15, 23, 69, 20, 12, 66, 17, 17, 9, 71, 22, 22, 14, 68, 19, 19, 11, 65, 73, 11, 16, 24, 16, 70, 8, 21, 21, 59, 13, 67, 75, 18, 18, 56, 26, 64, 72, 45, 10, 23, 15, 23, 61, 31, 69, 31, 77, 20, 20, 28, 58, 28, 12, 66, 74, 74, 17

Offset: 1

Jimin Park, Oct 16 2009

nonn

A given term k appears A131450(k) times. - Flávio V. Fernandes, Mar 13 2022

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to sequences related to the 3x+1 problem

Cf. A131450, A006577, A075680.

A006370 := proc(n) if type(n,'even') then n/2; else 3*n+1 ; end if; end proc:
A006577 := proc(n) a := 0 ; x := n ; while x > 1 do x := A006370(x) ; a := a+1 ; end do; a ; end proc:
A006667 := proc(n) a := 0 ; x := n ; while x > 1 do if type(x,'even') then x := x/2 ; else x := 3*x+1 ; a := a+1 ; end if; end do; a ; end proc:
A075680 := proc(n) A006667(2*n-1) ; end proc:
A166549 := proc(n) A006577(2*n-1)-A075680(n) ; end: seq(A166549(n),n=1..120) ; # R. J. Mathar, Oct 18 2009
# second Maple program:
b:= proc(n) option remember; `if`(n=1, 0,
      1+b(`if`(n::even, n/2, (3*n+1)/2)))
    end:
a:= n-> b(2*n-1):
seq(a(n), n=1..75);  # Alois P. Heinz, Mar 14 2022

b[n_] := b[n] = If[n == 1, 0, 1 + b[If[EvenQ[n], n/2, (3n+1)/2]]];
a[n_] := b[2n-1];
Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Apr 22 2022, after Alois P. Heinz *)

a(n) = A006577(2n-1) - A075680(n).

Edited and extended by R. J. Mathar, Oct 18 2009

cp's OEIS Frontend

User: Jimin Park

A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

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A378126 Array read by antidiagonals: T(n, m) is the maximal size of partitions of (n, m) into sums of distinct pairs of nonnegative integers, excluding (0, 0).

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A193865 a(n) = (A187108(n+1)-A187108(n))/2.

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AS3

A187108 Odd numbers not in the trajectory of a smaller number under the Collatz (3x+1) iteration.

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A166549 The number of halving steps of the Collatz 3x+1 map to reach 1 starting from 2n-1.

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