A332904 -id:A332904 - cp's OEIS frontend

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.

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

A332809 Number of distinct integers encountered on possible paths from n to 1 when iterating the nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 4, 6, 6, 7, 7, 8, 9, 10, 5, 6, 9, 10, 8, 12, 10, 11, 9, 9, 11, 10, 12, 13, 14, 15, 6, 17, 8, 18, 12, 13, 14, 15, 10, 11, 17, 18, 13, 18, 15, 16, 11, 18, 12, 14, 14, 15, 14, 16, 15, 17, 17, 18, 18, 19, 20, 20, 7, 22, 23, 24, 10, 26, 24, 25, 15, 16, 17, 21, 18, 30, 20, 21, 12, 15, 14, 15, 22, 16, 24, 25, 16

Offset: 1

Antti Karttunen, Apr 04 2020

The count includes also n itself, and the final 1 when it is distinct from n.

a(n) >= A000005(n) because all divisors of n can be found in the union of those paths. - Antti Karttunen, Apr 19 2020

			a(1): {1}, therefore a(1) = 1;
a(6): we have two alternative paths: {6, 4, 2, 1} or {6, 3, 2, 1}, with numbers [1, 2, 3, 4, 6] present, therefore a(6) = 5;
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) = 7;
a(14): we have five alternative paths: {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}, with numbers [1, 2, 3, 4, 6, 7, 8, 12, 14] present in at least one of the paths, therefore a(14) = 9.

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

Cf. A064097, A332810, A333123, A334230, A334231, A333786 (first occurrence of each n), A334112.
Partial sums of A332902.
See A332904 for the sum.

a[n_] := Block[{lst = {{n}}}, While[lst[[-1]] != {1}, lst = Join[ lst, {Union[ Flatten[# - #/(First@# & /@ FactorInteger@#) & /@ lst[[-1]]]]}]]; Length@ Union@ Flatten@ lst]; Array[a, 75] (* Robert G. Wilson v, Apr 06 2020 *)

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];

from sympy import factorint
from functools import cache
@cache
def b(n): return {n}.union(*(b(n - n//p) for p in factorint(n)))
def a(n): return len(b(n))
print([a(n) for n in range(1, 89)]) # Michael S. Branicky, Aug 13 2022

a(p) = 1 + a(p-1) for all primes p.
a(n) = n - A332810(n).
a(n) = A334112(n) + A000005(n). - Antti Karttunen, May 09 2020

A332994 a(1) = 1, for n > 1, a(n) = n + a(A052126(n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 9, 8, 15, 13, 13, 12, 19, 14, 17, 19, 31, 18, 27, 20, 27, 25, 25, 24, 39, 31, 29, 40, 35, 30, 39, 32, 63, 37, 37, 41, 55, 38, 41, 43, 55, 42, 51, 44, 51, 58, 49, 48, 79, 57, 63, 55, 59, 54, 81, 61, 71, 61, 61, 60, 79, 62, 65, 76, 127, 71, 75, 68, 75, 73, 83, 72, 111, 74, 77, 94, 83, 85, 87, 80, 111, 121

Offset: 1

Antti Karttunen, Apr 04 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. A000203, A052126, A322382, A332904, A332993, A333784, A333791, A333794.

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));

a(1) = 1; and for n > 1, a(n) = n + a(A052126(n)).
a(n) = n + A322382(n).
a(n) = A332993(n) - A333791(n).
a(n) = A000203(n) - A333784(n).

A333794 a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 20, 15, 22, 25, 36, 27, 40, 41, 42, 31, 48, 45, 64, 51, 66, 73, 96, 55, 76, 81, 72, 83, 112, 85, 116, 63, 118, 97, 120, 91, 128, 129, 130, 103, 144, 133, 176, 147, 136, 193, 240, 111, 182, 153, 162, 163, 216, 145, 208, 167, 202, 225, 284, 171, 232, 233, 208, 127, 236, 237, 304, 195, 306, 241, 312, 183, 256, 257

Offset: 1

Antti Karttunen, Apr 05 2020

nonn

Conjecturally, also the largest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

			For n=119, the graph obtained is this:
              119
             _/\_
            /    \
          102    112
         _/|\_    | \_
       _/  |  \_  |   \_
      /    |    \ |     \
    51     68    96     56
    /|   _/ |   _/|   _/ |
   / | _/   | _/  | _/   |
  /  |/     |/    |/     |
(48) 34    64     48    28
     |\_    |    _/|   _/|
     |  \_  |  _/  | _/  |
     |    \_|_/    |/    |
    17     32     24    14
      \_    |    _/|   _/|
        \_  |  _/  | _/  |
          \_|_/    |/    |
           16      12    7
            |    _/|    _/
            |  _/  |  _/
            |_/    |_/
            8     _6
            |  __/ |
            |_/    |
            4      3
             \     /
              \_ _/
                2
                |
                1.
If we always subtract A052126(n) (i.e., n divided by its largest prime divisor), i.e., iterate with A171462 (starting from 119), we obtain 119-(119/17) = 112 -> 112-(112/7) -> 96-(96/3) -> 64-(64/2) -> 32-(32/2) -> 16-(16/2) -> 8-(8/2) -> 4-(4/2) -> 2-(2/2) -> 1, with sum 119+112+96+64+32+16+8+4+2+1 = 554, thus a(119) = 554. This happens also to be maximal sum of any path in above diagram.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Michael De Vlieger, Graph montage of k -> k - k/p, with prime p|k for 2 <= k <= 211, red line showing path of greatest sum, blue the path of least sum (cf. A333790), and purple where the two paths coincide, with other paths in gray.

Cf. A052126, A073934, A171462, A332994, A333790, A333793.

Array[Total@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # > 1 &] &, 74] (* Michael De Vlieger, Apr 14 2020 *)

A333794(n) = if(1==n,n,n + A333794(n-(n/vecmax(factor(n)[, 1]))));

a(1) = 1; and for n > 1, a(n) = n + a(A171462(n)) = n + a(n-A052126(n)).
a(n) = A073934(n) + A333793(n).
a(n) = n + Max a(n - n/p), for p prime and dividing n. [Conjectured, holds at least up to n=2^24]
For all n >= 1, A333790(n) <= a(n) <= A332904(n).
For all n >= 1, a(n) >= A332993(n). [Apparently, have to check!]

A333000 Sum of integers (with multiplicity) encountered on all 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

1, 3, 6, 7, 12, 25, 39, 15, 43, 47, 69, 76, 115, 185, 198, 31, 48, 209, 304, 138, 604, 317, 432, 203, 213, 500, 344, 640, 901, 899, 1271, 63, 1777, 179, 2274, 736, 1069, 1572, 1860, 361, 525, 3156, 4360, 1074, 2580, 2150, 2808, 506, 4528, 924, 1042, 1630, 2266, 1836, 2878, 1930, 5004, 4165, 5522, 3026, 4307, 6343, 7638, 127, 6801

Offset: 1

Antti Karttunen, Apr 06 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}, therefore a(12) = (12+8+4+2+1) + (12+6+4+2+1) + (12+6+3+2+1) = 27+25+24 = 76
For n=15 we have five alternative paths from 15 to 1 (illustrated below): therefore a(15) = (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) = 37+40+42+40+39 = 198.
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.

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

Cf. A332904, A333001, A333123.

up_to = 20000;
A333000list(up_to) = { my(u=vector(up_to), v=vector(up_to)); u[1] = v[1] = 1; for(n=2,up_to, u[n] = vecsum(apply(p -> u[n-n/p], factor(n)[, 1]~)); v[n] = (u[n]*n)+vecsum(apply(p -> v[n-n/p], factor(n)[, 1]~))); (v); };
v333000 = A333000list(up_to);
A333000(n) = v333000[n];

a(n) = n*A333123(n) + Sum_{p prime and dividing n} a(n - n/p).

For all n, a(n) >= A332904(n).

A333790 Smallest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

Original entry on oeis.org

1, 3, 6, 7, 12, 12, 19, 15, 21, 22, 33, 24, 37, 33, 37, 31, 48, 39, 58, 42, 54, 55, 78, 48, 67, 63, 66, 61, 90, 67, 98, 63, 88, 82, 96, 75, 112, 96, 102, 82, 123, 96, 139, 99, 112, 124, 171, 96, 145, 117, 133, 115, 168, 120, 154, 117, 153, 148, 207, 127, 188, 160, 159, 127, 180, 154, 221, 150, 193, 166, 237, 147, 220, 186, 192, 172, 231

Offset: 1

Antti Karttunen, Apr 06 2020

nonn

Note that although in many cases a simple heuristics of always subtracting the largest proper divisor (i.e., iterating with A060681) gives the path with the minimal sum, this does not hold for the following numbers 119, 143, 187, 209, 221, ..., A333789, on which this sequence differs from A073934.

			For n=119, the graph obtained is this:
              119
             _/\_
            /    \
          102    112
         _/|\_    | \_
       _/  |  \_  |   \_
      /    |    \ |     \
    51     68    96     56
    /|   _/ |   _/|   _/ |
   / | _/   | _/  | _/   |
  /  |/     |/    |/     |
(48) 34    64     48    28
     |\_    |    _/|   _/|
     |  \_  |  _/  | _/  |
     |    \_|_/    |/    |
    17     32     24    14
      \_    |    _/|   _/|
        \_  |  _/  | _/  |
          \_|_/    |/    |
           16      12    7
            |    _/|    _/
            |  _/  |  _/
            |_/    |_/
            8     _6
            |  __/ |
            |_/    |
            4      3
             \     /
              \_ _/
                2
                |
                1.
By choosing the path that follows the right edge of the above diagram, we obtain the smallest sum for any such path that goes from 119 to 1, thus a(119) = 119+112+56+28+14+7+6+3+2+1 = 348.
Note that if we always subtracted the largest proper divisor (A032742), i.e., iterated with A060681 (starting from 119), we would obtain 119-(119/7) = 102 -> 102-(102/2) -> 51-(51/3) -> 34-(34/2) -> 17-(17/17) -> 16-(16/2) -> 8-(8/2) -> 4-(4/2) -> 2-(2/2) -> 1, with sum 119+102+51+34+17+16+8+4+2+1 = 354 = A073934(119), which is NOT minimal sum in this case.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Michael De Vlieger, Graph montage of k -> k - k/p, with prime p|k for 2 <= k <= 121, red line showing path of least sum.

Cf. A032742, A060681, A332904, A333000, A333001, A333123, A333794.

Differs from A073934 for the first time at n=119, where a(119) = 348, while A073934(119) = 354. (See A333789).

Min@ Map[Total, #] & /@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 76]   (* Michael De Vlieger, Apr 14 2020 *)

up_to = 65537; \\ 2^20;
A333790list(up_to) = { my(v=vector(up_to)); v[1] = 1; for(n=2, up_to, v[n] = n+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~))); (v); };
v333790 = A333790list(up_to);
A333790(n) = v333790[n];

a(n) = n + Min a(n - n/p), for p prime and dividing n.

For n >= 1, a(n) <= A333794(n) <= A332904(n), a(n) <= A333001(n).

A333793 a(n) = A333794(n) - A073934(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 3, 3, 3, 3, 8, 5, 0, 0, 6, 6, 9, 12, 18, 18, 7, 9, 18, 6, 22, 22, 18, 18, 0, 30, 15, 24, 16, 16, 33, 28, 21, 21, 37, 37, 48, 24, 69, 69, 15, 37, 36, 29, 48, 48, 25, 54, 50, 49, 77, 77, 44, 44, 73, 49, 0, 56, 83, 83, 45, 113, 75, 75, 36, 36, 71, 54, 87, 87, 81, 81, 45, 25, 84, 84, 87, 57, 128, 119, 108, 108, 71

Offset: 1

Antti Karttunen, Apr 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079, A019434, A032742, A052126, A073934, A332904, A333123, A333794.

Partial sums of A333792.

A073934(n) = if(1==n,n,n + A073934(n-(n/vecmin(factor(n)[,1]))));
A333794(n) = if(1==n,n,n + A333794(n-(n/vecmax(factor(n)[, 1]))));
A333793(n) = (A333794(n)-A073934(n));

a(n) = A333794(n) - A073934(n).
a(p) = a(p-1), for all primes p.
a(A000079(n)) = a(A019434(n)) = 0, for all applicable n.

cp's OEIS Frontend

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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A332809 Number of distinct integers encountered on possible paths from n to 1 when iterating the nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

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A332994 a(1) = 1, for n > 1, a(n) = n + a(A052126(n)).

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A333794 a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).

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A333000 Sum of integers (with multiplicity) encountered on all 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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A333790 Smallest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

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A333793 a(n) = A333794(n) - A073934(n).

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