A064097 -id:A064097 - cp's OEIS frontend

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.

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).

A049115 a(n) is the number of iterations of the Euler phi function needed to reach a power of 2, when starting from n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 1, 0, 1, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 0, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 2, 2, 3, 4, 1, 3, 2, 1, 2, 3, 3, 2, 2, 3, 3, 4, 1, 2, 2, 3, 0, 2, 2, 3, 1, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 1, 4, 2, 3, 2, 1, 3, 3, 2, 3, 2, 3, 3, 2, 4, 3, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2

Offset: 1

Labos Elemer

nonn

a(n) = A227944(n) if n is not a power of 2. - Eric M. Schmidt, Oct 13 2013

			If n is a power of 2, then a(n)=0 by definition. If n = 59049, then by iterating with phi, we get 59049 -> 39366 -> 13122 -> 4374 -> 1458 -> 486 -> 162 -> 54 -> 18 -> 6 -> 2 -> 1. It took ten steps to reach the first power of 2 (2 in this case), so a(59049) = 10.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A003434, A049113, A057716, A209229, A227944.

Cf. also A064097, A064415, A329697.

Table[If[IntegerQ@ Log2@ n, 0, -1 + Length@ NestWhileList[EulerPhi, n, ! IntegerQ@ Log2@ # &]], {n, 105}] (* Michael De Vlieger, Aug 01 2017 *)

A049115(n) = if(!bitand(n,n-1),0,1+A049115(eulerphi(n))); \\ Antti Karttunen, Aug 28 2021

The smallest x so that Nest[ EulerPhi, n, x ] = 2^w is just achieved.
From Antti Karttunen, Aug 28 2021: (Start)
If A209229(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A000010(n)).
a(n) <= A003434(n) and a(n) <= A329697(n) for all n.
(End)

Definition corrected and simplified, example corrected by Antti Karttunen, Aug 28 2021

A334144 Consider the mapping k -> (k - (k/p)), where prime p | k. a(n) = maximum distinct terms at any position j among the various paths to 1.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Peter Kagey, Antti Karttunen, Apr 15 2020

nonn

Let i = A064097(n) be the common path length and let 1 <= j <= i. Given a path P, we find for any j relatively few distinct values. Regarding a common path length i, see A333123 comment 2, and proof at A064097.

Maximum term in row n of A334184.

			For n=15, the paths are shown vertically at left, and the graph obtained appears at right:
  15   15   15   15   15  =>         15
   |    |    |    |    |            _/ \_
   |    |    |    |    |           /     \
  10   10   12   12   12  =>     10       12
   |    |    |    |    |         | \_   _/ |
   |    |    |    |    |         |   \ /   |
   5    8    6    6    8  =>     5    8    6
   |    |    |    |    |          \_  |  _/|
   |    |    |    |    |            \_|_/  |
   4    4    3    4    4  =>          4    3
   |    |    |    |    |              |  _/
   |    |    |    |    |              |_/
   2    2    2    2    2  =>          2
   |    |    |    |    |              |
   |    |    |    |    |              |
   1    1    1    1    1  =>          1
Because the maximum number of distinct terms in any row is 3, a(15) = 3.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Math.StackExchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
Richard Stanley, MIT OpenCourseWare, Lecture Notes in Algebraic Combinatorics: The Sperner Property
Wikipedia, Sperner property of a partially ordered set

Cf. A064097, A332992, A332999, A333123, A334111, A334184.

Cf. also A096825.

Max[Length@ Union@ # & /@ Transpose@ #] & /@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 105]
(* Second program: *)
g[n_] := Block[{lst = {{n}}}, While[lst[[-1]] != {1}, lst = Join[lst, {Union@ Flatten[# - #/(First@ # & /@ FactorInteger@ #) & /@ lst[[-1]] ]}]]; Max[Length /@ lst]]; Array[g, 105] (* Robert G. Wilson v, May 08 2020 *)

A175125 a(n) is the number of numbers m such that the number of iterations of r -> r - (largest divisor d < r) needed to reach 1 starting at r = m is equal to n.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 13, 23, 36, 65, 100, 175, 275, 468, 760, 1266, 2077, 3454, 5690, 9449, 15547, 25709, 42459, 70181, 115947, 191509, 316570, 523087, 864406, 1428174, 2359266

Offset: 0

Jaroslav Krizek, Feb 15 2010

			Example (a(4)=5): There are five numbers (7,9,10,12,16) with 4 steps of defined iteration: 7-1=6, 6-3=3, 3-1=2, 2-1=1; 9-3=6, 6-3=3, 3-1=2, 2-1=1; 10-5=5, 5-1=4, 4-2=2, 2-1=1; 12-6=6, 6-3=3, 3-1=2, 2-1=1; 16-8=8, 8-4=4, 4-2=2, 2-1=1.

Cf. A000079, A064097, A105017.

a064097(n)={my(k=1,d=divisors(n),r=n-d[#d-1]);while(r>1,d=divisors(r);r-=d[#d-1];k++);k};
my(c=vector(50));for(k=2,2^20,j=a064097(k);c[j]++);concat([1],c[1..20]) \\ Hugo Pfoertner, Mar 23 2020

a(8)-a(27) from Hugo Pfoertner, Mar 23 2020

a(28)-a(30) from Robert G. Wilson v, Apr 05 2020

A334200 Fully additive with a(n) = n-1 for n <= 3, and a(p) = 1 + a(A048673(p)) when p is prime > 3 and a(n*m) = a(n) + a(m) when m,n > 1.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 5, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 6, 7, 7, 7, 7, 6, 8, 7, 7, 7, 7, 7, 8, 7, 7, 6, 7, 7, 7, 7, 8, 6, 8, 8, 7, 7, 8, 8, 8, 7, 7, 8, 8, 7, 9, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 9, 8, 9, 8, 8, 8, 8, 7, 8, 9, 9, 8, 8, 8, 8, 8, 9

Offset: 1

Antti Karttunen, May 13 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A048673, A334199.

Cf. also A064097, A334206.

A334200(n) = if(n<=3, n-1, if(isprime(n), 1+A334200((1+nextprime(1+n))/2), my(f=factor(n)); (apply(A334200, f[, 1])~ * f[, 2])));

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.

A352957 Triangle read by rows: Row n is the lexicographically earliest strictly monotonic completely additive sequence of length n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 2, 3, 4, 0, 2, 3, 4, 5, 0, 3, 5, 6, 7, 8, 0, 3, 5, 6, 7, 8, 9, 0, 4, 6, 8, 9, 10, 11, 12, 0, 5, 8, 10, 11, 13, 14, 15, 16, 0, 5, 8, 10, 12, 13, 14, 15, 16, 17, 0, 5, 8, 10, 12, 13, 14, 15, 16, 17, 18, 0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25

Offset: 1

Peter Munn, Apr 11 2022

Each sequence consists of nonnegative integers indexed from 1.
Note in particular in the formula section, the lower bound, floor(n/k), for first differences between terms in a row. This follows (using the additive property) from the strict monotonicity of floor(n/k)+1 consecutive terms near the end of the row.
For any k, with increasing length n >= k, the first k terms of the sequences approach similarity with a real-valued logarithmic function defined on the integers. For example, the asymptote of T(n,3)/T(n,2) is log(3)/log(2), A020857.

			(For row 4.) A completely additive sequence requires T(4,1) = 0. Strict monotonicity requires T(4,4) > T(4,3) > T(4,2). So T(4,4) >= T(4,2) + 2. Using the additivity this becomes T(4,2) + T(4,2) >= T(4,2) + T(4,1) + 2. Subtracting T(4,2) and substituting 0 for T(4,1) we get T(4,2) >= 2. So from T(4,4) > T(4,3) > T(4,2), we see T(4,3) >= 3, T(4,4) >= 4. So row 4 = (0, 2, 3, 4) as it is strictly monotonic and completely additive and from the preceding arguments is seen to be the lexicographically earliest such.
Triangle starts:
0;
0, 1;
0, 1,  2;
0, 2,  3,  4;
0, 2,  3,  4,  5;
0, 3,  5,  6,  7,  8;
0, 3,  5,  6,  7,  8,  9;
0, 4,  6,  8,  9, 10, 11, 12;
0, 5,  8, 10, 11, 13, 14, 15, 16;
0, 5,  8, 10, 12, 13, 14, 15, 16, 17;
0, 5,  8, 10, 12, 13, 14, 15, 16, 17, 18;
0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25;
0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25, 26;
0, 7, 11, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27;
0, 8, 13, 16, 19, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32;
0, 9, 14, 18, 21, 23, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36;

Peter Munn, Rows n = 1..141, flattened
Encyclopedia of Mathematics, Additive arithmetic function.
Peter Munn, PARI program

Cf. A020857.
Completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1): A064097, A344443, A344444; and for functions of earlier terms, see A334200.
For completely additive sequences with primes p mapped to a function of p, see A001414.
For completely additive sequences with prime(k) mapped to a function of k, see A104244.
For completely additive sequences where some primes are mapped to 1, the rest to 0 (notably, some ruler functions) see the cross-references in A249344.

The definition specifies: T(n,j*k) = T(n,j) + T(n,k); for k > 1, T(n,k) > T(n,k-1).
T(n,1) = 0, otherwise T(n,k) >= T(n,k-1) + floor(n/k).
For prime p, T(p,p) = T(p-1,p-1) + 1, otherwise T(p,k) = T(p-1,k).
T(n,2) >= 2*floor(n/4) + floor(n/9).
T(n,3) >= ceiling( (3*T(n,2) + floor(n/9)) / 2).
T(11,k) = A344443(k).
For k <> 13, T(23,k) = A344444(k).

A117618 Least number with complexity height of n, under integer complexity A005245.

Original entry on oeis.org

1, 6, 7, 10, 22, 683

Offset: 1

Jonathan Vos Post, Apr 07 2006

Consider the recursion: A005245(n), A005245(A005245(n)), A005245(A005245(A005245(n))), ... which we know is finite before reaching a fixed point, as A005245(n) <= n. The number of steps needed to reach such a fixed point is the complexity height of n (with respect to the A005245 measure of complexity, there being others in the OEIS).

a(7) >= 872573642639 = A005520(89). - David A. Corneth, May 06 2024

			a(1) = 1 because the A005245 complexity of 1 is 1, already giving a fixed point.
a(2) = 6 because it is the smallest x such that A005245(x) =/= x and A005245(x) = A005245(A005245(x)).
a(3) = 7 because 7 is the least number x with complexity 6, thus taking a further step of recursion to reach a fixed point.
a(4) = 10 because 10 is the least number with complexity 7.
a(5) = 22 because 22 is the least number with complexity 10.
a(6) = 683 because 683 is the least number with complexity 22.
a(7) = the least number with complexity 683.

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971.
R. K. Guy, Unsolved Problems in Number Theory, Sect. F26.

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
E. Pegg, Jr., Integer Complexity.
Eric Weisstein's World of Mathematics, Integer Complexity.
Index to sequences related to the complexity of n

Cf. A003037, A003313, A005245, A005421, A005520, A025280, A061373, A064097, A076091, A076142.

a(n) = least k such that A005245^(n)(k) = A005245^(n-1)(k) but (if n>1) A005245^(n-1)(k) != A005245^(n-2)(k), where ^ denotes repeated application.

For n >= 3, a(n) = A005520(a(n-1)). - Max Alekseyev, May 06 2024

a(2)=6 inserted by Giovanni Resta, Jun 15 2016

Edited by Max Alekseyev, May 06 2024

A191614 The lexicographically earliest sequence such that a(n) - a(n-1) is the largest proper divisor of a(n).

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 21, 42, 43, 86, 129, 258, 301, 602, 903, 1806, 1849, 3698, 5547, 11094, 12943, 25886, 38829, 77658, 77659, 155318, 232977, 465954, 543613, 1087226, 1630839, 3261678, 3339337, 6678674, 10018011, 20036022, 23375359, 46750718, 70126077, 140252154

Offset: 1

Jaroslav Krizek, Jun 09 2011

nonn

In contrast, A000079(n) is the lexicographically *largest* sequence such that a(n) - a(n-1) is the largest proper divisor of a(n).

Sequence is infinite because A060681 is surjective.

Cf. A105017, A064097, A175125, A060681.

A191614 := proc(n) option remember; if n = 1 then 1; else for m from 1 do if m-A032742(m) = procname(n-1) then return m; end if; end do: end if; end proc: # R. J. Mathar, Jun 13 2011

p=0; for (v=1, 140252154, if (v%(v-p)==0 && (p==0 || (d=divisors(v))[#d-1]==v-p), print1 (p=v ", "))) \\ Rémy Sigrist, Apr 24 2021

a(n) is the smallest number m such that m - A032742(m) = a(n-1), n > 1.

More terms from Rémy Sigrist, Apr 24 2021

cp's OEIS Frontend

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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A049115 a(n) is the number of iterations of the Euler phi function needed to reach a power of 2, when starting from n.

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A334144 Consider the mapping k -> (k - (k/p)), where prime p | k. a(n) = maximum distinct terms at any position j among the various paths to 1.

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A175125 a(n) is the number of numbers m such that the number of iterations of r -> r - (largest divisor d < r) needed to reach 1 starting at r = m is equal to n.

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A334200 Fully additive with a(n) = n-1 for n <= 3, and a(p) = 1 + a(A048673(p)) when p is prime > 3 and a(n*m) = a(n) + a(m) when m,n > 1.

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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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A352957 Triangle read by rows: Row n is the lexicographically earliest strictly monotonic completely additive sequence of length n.

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