A332809 -id:A332809 - cp's OEIS frontend

A332902 a(1) = 1, then after the first differences of A332809.

1, 1, 1, 0, 1, 1, 1, -2, 2, 0, 1, 0, 1, 1, 1, -5, 1, 3, 1, -2, 4, -2, 1, -2, 0, 2, -1, 2, 1, 1, 1, -9, 11, -9, 10, -6, 1, 1, 1, -5, 1, 6, 1, -5, 5, -3, 1, -5, 7, -6, 2, 0, 1, -1, 2, -1, 2, 0, 1, 0, 1, 1, 0, -13, 15, 1, 1, -14, 16, -2, 1, -10, 1, 1, 4, -3, 12, -10, 1, -9, 3, -1, 1, 7, -6, 8, 1, -9, 1, 7, 4, -9, 8, -6, 7, -15, 1, 10

Offset: 1

Antti Karttunen, Apr 04 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..10200
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A332809, A332903.

up_to = 105;
A332809list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(length,v); }
v332809 = A332809list(up_to);
A332809(n) = v332809[n];
A332902(n) = if(1==n,n,(A332809(n)-A332809(n-1)));

a(1) = 1, and for n > 1, a(n) = A332809(n) - A332809(n-1).

a(p) = 1 for all primes p.

A333786 a(n) is the first occurrence of n in A332809.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 23, 21, 29, 30, 31, 47, 33, 35, 61, 62, 75, 65, 66, 67, 71, 69, 93, 91, 131, 77, 134, 142, 133, 105, 139, 143, 141, 265, 154, 195, 366, 201, 266, 210, 161, 209, 213, 215, 355, 217, 335, 377, 299, 315, 319, 231, 322, 403, 419, 417, 431, 585, 329, 435, 429, 497, 638, 437, 631, 795

Offset: 1

Michael De Vlieger, Peter Kagey, Antti Karttunen, Robert G. Wilson v, May 09 2020

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..15966 (first 1000 terms from Peter Kagey)

Cf. A332809 (a left inverse).

a[n_] := Block[{lst = {{n}}}, While[ lst[[-1]] != {1}, lst = Join[lst, {Union[ Flatten[# - #/(First@# & /@ FactorInteger@#) & /@ lst[[-1]]]]}]]; Length@Flatten@lst]; t[] := 0; k = 1; While[k < 100001, b = a@k; If[ t[b] == 0, t[b] = k]; k++]; t@# & /@ Range@ 100 (* _Robert G. Wilson v, May 08 2020 *)

search_up_to = 20000;
A332809list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(length,v); }
v332809 = A332809list(search_up_to);
A332809(n) = v332809[n];
A333786list(search_up_to) = { my(focs=Map(),t); for(n=1,search_up_to,t = A332809(n); if(!mapisdefined(focs,t),mapput(focs,t,n))); my(lista=List([]),v); for(u=1,oo,if(!mapisdefined(focs,u,&v), return(Vec(lista)), listput(lista,v))); };
v333786 = A333786list(search_up_to);
A333786(n) = v333786[n]; \\ Antti Karttunen, May 09 2020

from sympy import factorint
from functools import cache
from itertools import count, islice
@cache
def b(n): return {n}.union(*(b(n - n//p) for p in factorint(n)))
def A332809(n): return len(b(n))
def agen():
    adict, n = dict(), 1
    for k in count(1):
        v = A332809(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Aug 13 2022

A332809(a(n)) = n.

A334112 a(n) = A332809(n) - A000005(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 0, 3, 2, 5, 1, 6, 5, 6, 0, 4, 3, 8, 2, 8, 6, 9, 1, 6, 7, 6, 6, 11, 6, 13, 0, 13, 4, 14, 3, 11, 10, 11, 2, 9, 9, 16, 7, 12, 11, 14, 1, 15, 6, 10, 8, 13, 6, 12, 7, 13, 13, 16, 6, 17, 16, 14, 0, 18, 15, 22, 4, 22, 16, 23, 3, 14, 13, 15, 12, 26, 12, 19, 2, 10, 10, 13, 10, 12, 20, 21, 8, 15, 12, 24, 13, 23

Offset: 1

Antti Karttunen, May 09 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A000079 (positions of zeros), A007283, A332809.

Array[Block[{w = {{#}}}, While[w[[-1]] != {1}, w = Join[w, {Union@ Flatten[# - #/FactorInteger[#][[All, 1]] & /@ w[[-1]] ]}]]; Length@ Union@ Flatten@ w - DivisorSigma[0, #]] &, 93] (* Michael De Vlieger, May 09 2020, after Robert G. Wilson v at A332809 *)

up_to = 20000;
A332809list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(length,v); }
v332809 = A332809list(up_to);
A332809(n) = v332809[n];
A334112(n) = (A332809(n) - numdiv(n));

a(n) = A332809(n) - A000005(n).

a(2^n) = 0 for all n >= 0.

A333123 Consider the mapping k -> (k - (k/p)), where p is any of k's prime factors. a(n) is the number of different possible paths from n to 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 5, 5, 1, 1, 5, 5, 3, 10, 5, 5, 4, 3, 7, 5, 9, 9, 12, 12, 1, 17, 2, 21, 9, 9, 14, 16, 4, 4, 28, 28, 9, 21, 14, 14, 5, 28, 7, 7, 12, 12, 14, 16, 14, 28, 23, 23, 21, 21, 33, 42, 1, 33, 47, 47, 3, 61, 56, 56, 14, 14, 23, 28, 28, 103, 42, 42, 5

Offset: 1

Ali Sada and Robert G. Wilson v, Mar 09 2020

The iteration always terminates at 1, regardless of the prime factor chosen at each step.
Although there may exist multiple paths to 1, their path lengths (A064097) are the same! See A064097 for a proof. Note that this behavior does not hold if we allow any divisor of k.
First occurrence of k or 0 if no such value exists: 1, 6, 12, 24, 14, 96, 26, 85, 28, 21, 578, 30, 194, 38, 164, 39, 33, 104, 1538, 112, 35, 328, 58, 166, ..., .
Records: 1, 2, 3, 5, 10, 12, 17, 21, 28, 33, 42, 47, 61, 103, 168, ..., .
Record indices: 1, 6, 12, 14, 21, 30, 33, 35, 42, 62, 63, 66, 69, ..., .
When viewed as a graded poset, the paths of the said graph are the chains of the corresponding poset. This poset is also a lattice (see Ewan Delanoy's answer to Peter Kagey's question at the Mathematics Stack Exchange link). - Antti Karttunen, May 09 2020

			a(1): {1}, therefore a(1) = 1;
a(6): {6, 4, 2, 1} or {6, 3, 2, 1}, therefore a(6) = 2;
a(12): {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, therefore a(12) = 3;
a(14): {14, 12, 8, 4, 2, 1}, {14, 12, 6, 4, 2, 1}, {14, 12, 6, 3, 2, 1}, {14, 7, 6, 4, 2, 1} or {14, 7, 6, 3, 2, 1}, therefore a(14) = 5.
From _Antti Karttunen_, Apr 05 2020: (Start)
For n=15 we have five alternative paths from 15 to 1: {15, 10, 5, 4, 2, 1}, {15, 10, 8, 4, 2, 1}, {15, 12, 8, 4, 2, 1},  {15, 12, 6, 4, 2, 1},  {15, 12, 6, 3, 2, 1}, therefore a(15) = 5. These form a graph illustrated below:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \_   |  __/|
     \__|_/   |
        4     3
         \   /
          \ /
           2
           |
           1
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Peter Kagey, Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?

Cf. A064097, A332809 (size of the lattice), A332810.
Cf. A332904 (sum of distinct integers present in such a graph/lattice), A333000 (sum over all paths), A333001, A333785.
Cf. A332992 (max. outdegree), A332999 (max. indegree), A334144 (max. rank level).
Cf. A334230, A334231 (meet and join).
Partial sums of A332903.
Cf. also tables A334111, A334184.

a[n_] := Sum[a[n - n/p], {p, First@# & /@ FactorInteger@n}]; a[1] = 1; (* after PARI coding by Rémy Sigrist *) Array[a, 70]
(* view the various paths *)
f[n_] := Block[{i, j, k, p, q, mtx = {{n}}}, Label[start]; If[mtx[[1, -1]] != 1, j = Length@ mtx;  While[j > 0, k = mtx[[j, -1]]; p = First@# & /@ FactorInteger@k; q = k - k/# & /@ p; pl = Length@p; If[pl > 1, Do[mtx = Insert[mtx, mtx[[j]], j], {pl - 1}]]; i = 1;  While[i < 1 + pl, mtx[[j + i - 1]] = Join[mtx[[j + i - 1]], {q[[i]]}]; i++]; j--]; Goto[start], mtx]]

for (n=1, #a=vector(80), print1 (a[n]=if (n==1, 1, vecsum(apply(p -> a[n-n/p], factor(n)[,1]~)))", ")) \\ Rémy Sigrist, Mar 11 2020

a(n) = 1 iff n is a power of two (A000079) or a Fermat Prime (A019434).
a(p) = a(p-1) if p is prime.
a(n) = Sum_{p prime and dividing n} a(n - n/p) for any n > 1. - Rémy Sigrist, Mar 11 2020

A334184 Irregular table read by rows: T(n,k) gives the number of values that can be reached after exactly k iterations of maps of the form (n - n/p) where p is a prime divisor of n. 0 <= k < A073933(n).

Original entry on oeis.org

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

Offset: 1

Peter Kagey, Apr 17 2020

Row lengths are given by A073933(n). Row sums are given by A332809(n). The maximum value in each row is given by A334144(n).
The n-th row consists of all 1's if and only if n is a power of two (A000079) or a Fermat prime (A019434).
Conjecture: rows are unimodal (increasing and then decreasing).
Not all rows are unimodal. Indices of rows that have terms that increase and decrease more than once are A334238. - Michael De Vlieger, Apr 18 2020

			For n = 15, the fifteenth row of this table is [1,2,3,2,1,1] because there is one value (15 itself) that can be reached with zero iterations of (n - n/p) maps, two values (10 and 12) that can be reached after one iteration, three values (5, 8, and 6) that can be reached after two iterations, and so on.
      15
     _/ \_
    /     \
  10       12
  | \_   _/ |
  |   \ /   |
  5    8    6
   \_  |  _/|
     \_|_/  |
       4    3
       |  _/
       |_/
       2
       |
       |
       1
Table begins:
1
1, 1
1, 1, 1
1, 1, 1
1, 1, 1, 1
1, 2, 1, 1
1, 1, 2, 1, 1
1, 1, 1, 1
1, 1, 2, 1, 1
1, 2, 1, 1, 1
1, 1, 2, 1, 1, 1
1, 2, 2, 1, 1
1, 1, 2, 2, 1, 1
1, 2, 2, 2, 1, 1
1, 2, 3, 2, 1, 1
1, 1, 1, 1, 1

Michael De Vlieger, Table of n, a(n) for n = 1..12386 (rows 1 <= n <= 1000, flattened)
Michael De Vlieger, Hasse diagrams showing rows n = {55, 63, 171, ...} that increase and decrease more than once.
Michael De Vlieger, Table of n, b(n) for n = 1..10000, encoding the running total of row n of this sequence as a binary number expressed decimally.

Cf. A001221, A073933, A332809, A333123, A334111, A334144, A334238.

Cf. A000079, A019434.

Table[Length@ Union@ # & /@ Transpose@ # &@ If[n == 1, {{1}}, NestWhile[If[Length[#] == 0, Map[{n, #} &, # - # /FactorInteger[#][[All, 1]] ], Union[Join @@ Map[Function[{w, n}, Map[Append[w, If[n == 0, 0, n - n/#]] &, FactorInteger[n][[All, 1]] ]] @@ {#, Last@ #} &, #]]] &, n, If[ListQ[#], AllTrue[#, Last[#] > 1 &], # > 1] &]], {n, 22}] // Flatten (* Michael De Vlieger, Apr 18 2020 *)

T(n,0) = T(n, A073933(n) - 2) = T(n, A073933(n) - 1) = 1.

T(n,1) = A001221(n) for n > 1.

A332904 Sum of distinct integers encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

Original entry on oeis.org

1, 3, 6, 7, 12, 16, 23, 15, 25, 30, 41, 36, 49, 57, 66, 31, 48, 63, 82, 66, 105, 99, 122, 76, 91, 115, 90, 125, 154, 156, 187, 63, 222, 114, 240, 139, 176, 196, 217, 138, 179, 251, 294, 215, 264, 284, 331, 156, 300, 213, 258, 247, 300, 220, 345, 261, 334, 348, 407, 336, 397, 429, 395, 127, 492, 512, 579, 246, 650, 546, 617, 291, 364

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

			a(12): we have three alternative paths: {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, with numbers [1, 2, 3, 4, 6, 8, 12] present, therefore a(12) = 1+2+3+4+6+8+12 = 36.
For n=15 we have five alternative paths from 15 to 1: {15, 10, 5, 4, 2, 1}, {15, 10, 8, 4, 2, 1}, {15, 12, 8, 4, 2, 1},  {15, 12, 6, 4, 2, 1},  {15, 12, 6, 3, 2, 1}. These form a lattice illustrated below:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1,
therefore a(15) = 1+2+3+4+5+6+8+10+12+15 = 66.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A064097, A073934, A332809, A332993, A332994, A333000, A333123.

Cf. A333790 (sum of the route with minimal sum), A333794.

Total /@ Nest[Function[{a, n}, Append[a, Union@ Flatten@ Table[Append[a[[n - n/p]], n], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{1}}, 72] (* Michael De Vlieger, Apr 15 2020 *)

up_to = 20000;
A332904list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(vecsum,v); }
v332904 = A332904list(up_to);
A332904(n) = v332904[n];

For all primes p, a(p) = a(p-1) + p.

For all n >= 1, A333000(n) >= a(n) >= A333794(n) >= A333790(n).

A332810 Number of integers in range 1..n that are not encountered on any of the possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p is any prime factor of k.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 4, 3, 4, 4, 5, 5, 5, 5, 11, 11, 9, 9, 12, 9, 12, 12, 15, 16, 15, 17, 16, 16, 16, 16, 26, 16, 26, 17, 24, 24, 24, 24, 30, 30, 25, 25, 31, 27, 31, 31, 37, 31, 38, 37, 38, 38, 40, 39, 41, 40, 41, 41, 42, 42, 42, 43, 57, 43, 43, 43, 58, 43, 46, 46, 57, 57, 57, 54, 58, 47, 58, 58, 68, 66, 68, 68, 62, 69, 62, 62, 72, 72

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

			a(12): we have three alternative paths: {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, with [5, 7, 9, 10, 11] being the only numbers in range 1..12 that do not occur in any of those paths, therefore a(12) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A064097, A332809, A333123.

up_to = 105;
A332809list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(length,v); }
v332809 = A332809list(up_to);
A332810(n) = (n-v332809[n]);

a(n) = n - A332809(n).

A332992 Maximum outdegree in the graph formed by a subset of numbers in range 1 .. n with edge relation k -> k - k/p, where p can be any of the prime factors of k.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 1, 3, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

Maximum number of distinct prime factors of any one integer encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p can be any of the prime factors of k.

			For n=15 we have five alternative paths from 15 to 1: {15, 10, 5, 4, 2, 1}, {15, 10, 8, 4, 2, 1}, {15, 12, 8, 4, 2, 1},  {15, 12, 6, 4, 2, 1},  {15, 12, 6, 3, 2, 1}. These form a lattice illustrated below:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1
With edges going from 15 towards 1, the maximum outdegree is 2, which occurs at nodes 15, 12, 10 and 6, therefore a(15) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A001221, A064097, A332809, A332999 (max. indegree), A333123, A334111, A334144, A334184.

Cf. A002110 (positions of records and the first occurrence of each n).

With[{s = Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 105]}, Array[If[# == 1, 0, Max@ Tally[#][[All, -1]] &@ Union[Join @@ Map[Partition[#, 2, 1] &, s[[#]] ]][[All, 1]] ] &, Length@ s]] (* Michael De Vlieger, May 02 2020 *)

up_to = 105;
A332992list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2,up_to, v[n] = max(omega(n),vecmax(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v332992 = A332992list(up_to);
A332992(n) = v332992[n];

a(n) = max(A001221(n), {Max a(n - n/p), for p prime and dividing n}).
For all odd primes p, a(p) = a(p-1).
For all n >= 0, a(A002110(n)) = n.

A334230 Triangle read by rows: T(n,k) gives the meet of n and k in the graded lattice of the positive integers defined by covering relations "n covers (n - n/p)" for all divisors p of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 4, 5, 1, 2, 3, 4, 4, 6, 1, 2, 3, 4, 4, 6, 7, 1, 2, 2, 4, 4, 4, 4, 8, 1, 2, 3, 4, 4, 6, 6, 4, 9, 1, 2, 2, 4, 5, 4, 4, 8, 4, 10, 1, 2, 2, 4, 5, 4, 4, 8, 4, 10, 11, 1, 2, 3, 4, 4, 6, 6, 8, 6, 8, 8, 12, 1, 2, 3, 4, 4, 6, 6, 8

Offset: 1

Peter Kagey, Antti Karttunen, and Michael De Vlieger, Apr 19 2020

Any row with prime index p is a copy of row p-1 followed by that prime p.

			The interval [1,15] illustrates that, for example, T(12, 10) = 8, T(12, 4) = T(5, 6) = 4, T(8, 3) = 2, etc.
      15
     _/ \_
    /     \
  10       12
  | \_   _/ |
  |   \ /   |
  5    8    6
   \_  |  _/|
     \_|_/  |
       4    3
       |  _/
       |_/
       2
       |
       |
       1
Triangle begins:
  n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
  ---+---------------------------------
   1 | 1
   2 | 1 2
   3 | 1 2 3
   4 | 1 2 2 4
   5 | 1 2 2 4 5
   6 | 1 2 3 4 4 6
   7 | 1 2 3 4 4 6 7
   8 | 1 2 2 4 4 4 4 8
   9 | 1 2 3 4 4 6 6 4 9
  10 | 1 2 2 4 5 4 4 8 4 10
  11 | 1 2 2 4 5 4 4 8 4 10 11
  12 | 1 2 3 4 4 6 6 8 6  8  8 12
  13 | 1 2 3 4 4 6 6 8 6  8  8 12 13
  14 | 1 2 3 4 4 6 7 8 6  8  8 12 12 14

Antti Karttunen, Table of n, a(n) for n = 1..10440; The first 144 rows, flattened
Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
Wikipedia, Semilattice

Cf. A332809, A333123, A334184, A334231.

\\ This just returns the largest (in a normal sense) number x from the intersection of the set of descendants of n and k:
up_to = 105;
buildWdescsets(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); (v); }
vdescsets = buildWdescsets(up_to);
A334230tr(n,k) = vecmax(setintersect(vdescsets[n],vdescsets[k]));
A334230list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A334230tr(n,k))); (v); };
v334230 = A334230list(up_to);
A334230(n) = v334230[n]; \\ Antti Karttunen, Apr 19 2020

T(n, k) = m*T(n/m, k/m) for m = gcd(n, k).

A334231 Triangle read by rows: T(n,k) gives the join of n and k in the graded lattice of the positive integers defined by covering relations "n covers (n - n/p)" for all divisors p of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 6, 4, 5, 5, 15, 5, 5, 6, 6, 6, 6, 15, 6, 7, 7, 7, 7, 35, 7, 7, 8, 8, 12, 8, 10, 12, 14, 8, 9, 9, 9, 9, 45, 9, 21, 18, 9, 10, 10, 15, 10, 10, 15, 35, 10, 45, 10, 11, 11, 33, 11, 11, 33, 77, 11, 99, 11, 11, 12, 12, 12, 12, 15, 12, 14, 12

Offset: 1

Peter Kagey, Apr 19 2020

The poset of the positive integers is defined by covering relations "n covers (n - n/p)" for all divisors p of n.

n appears A332809(n) times in row n.

			The interval [1,15] illustrates that, for example, T(12, 10) = T(6, 5) = 15, T(12, 4) = 12, T(8, 5) = 10, T(3, 1) = 3, etc.
      15
     _/ \_
    /     \
  10       12
  | \_   _/ |
  |   \ /   |
  5    8    6
   \_  |  _/|
     \_|_/  |
       4    3
       |  _/
       |_/
       2
       |
       |
       1
Triangle begins:
  n\k|  1  2  3  4  5  6  7  8  9 10  11 12 13 14
  ---+-------------------------------------------
   1 |  1
   2 |  2  2
   3 |  3  3  3
   4 |  4  4  6  4
   5 |  5  5 15  5  5
   6 |  6  6  6  6 15  6
   7 |  7  7  7  7 35  7  7
   8 |  8  8 12  8 10 12 14  8
   9 |  9  9  9  9 45  9 21 18  9
  10 | 10 10 15 10 10 15 35 10 45 10
  11 | 11 11 33 11 11 33 77 11 99 11  11
  12 | 12 12 12 12 15 12 14 12 18 15  33 12
  13 | 13 13 13 13 65 13 91 13 39 65 143 13 13
  14 | 14 14 14 14 35 14 14 14 21 35  77 14 91 14

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
Wikipedia, Semilattice

Cf. A051173, A332809, A333123, A334184, A334230.

\\ This just returns the least (in a normal sense) number x such that both n and k are in its set of descendants:
up_to = 105;
buildWdescsets(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); (v); }
vdescsets = buildWdescsets(100*up_to); \\ XXX - Think about a safe limit here!
A334231tr(n,k) = for(i=max(n,k),oo,if(setsearch(vdescsets[i],n)&&setsearch(vdescsets[i],k),return(i)));
A334231list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A334231tr(n,k))); (v); };
v334231 = A334231list(up_to);
A334231(n) = v334231[n]; \\ Antti Karttunen, Apr 19 2020

T(n,1) = T(n,n) = n. T(n, 2) = n for n >= 2.
T(x,y) <= lcm(x,y) for any x,y because x is in same chain with lcm(x,y), and y is in same chain with lcm(x,y).
Moreover, empirically it looks like T(x,y) divides lcm(x,y).

cp's OEIS Frontend

A332902 a(1) = 1, then after the first differences of A332809.

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A333786 a(n) is the first occurrence of n in A332809.

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A334112 a(n) = A332809(n) - A000005(n).

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A333123 Consider the mapping k -> (k - (k/p)), where p is any of k's prime factors. a(n) is the number of different possible paths from n to 1.

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A334184 Irregular table read by rows: T(n,k) gives the number of values that can be reached after exactly k iterations of maps of the form (n - n/p) where p is a prime divisor of n. 0 <= k < A073933(n).

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A332904 Sum of distinct integers encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

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A332810 Number of integers in range 1..n that are not encountered on any of the possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p is any prime factor of k.

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A332992 Maximum outdegree in the graph formed by a subset of numbers in range 1 .. n with edge relation k -> k - k/p, where p can be any of the prime factors of k.

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