A333790 -id:A333790 - cp's OEIS frontend

A333789 Numbers k for which A333790(k) < A073934(k).

119, 143, 187, 209, 221, 238, 239, 286, 319, 357, 374, 407, 418, 419, 429, 442, 443, 451, 476, 478, 479, 561, 572, 595, 627, 638, 663, 667, 671, 703, 713, 714, 715, 717, 748, 779, 803, 814, 833, 836, 838, 839, 851, 858, 859, 884, 886, 887, 902, 935, 943, 952, 953, 956, 957, 958, 979, 989, 1001, 1045, 1067, 1071, 1073, 1105, 1111, 1122

Offset: 1

Antti Karttunen, Apr 12 2020

nonn

Numbers n for which the {smallest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k} cannot be obtained by always selecting the smallest prime factor of k (A020639). See the example in A333790 how that simple heuristic fails when starting from k=119.

Antti Karttunen, Table of n, a(n) for n = 1..36576; all terms <= 2^17

Cf. A020639, A060681, A073934, A333790.

Block[{a, b, nn = 1122}, a = 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}}}, nn]; b = Array[If[# == 1, 1, Total@ NestWhileList[If[PrimeQ@ #, # - 1, # - #/FactorInteger[#][[1, 1]] ] &, #, # > 1 &]] &, nn]; Select[Range@ nn, a[[#]] < b[[#]] &]] (* Michael De Vlieger, Apr 15 2020 *)

search_up_to = 2^17;
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(search_up_to);
A333790(n) = v333790[n];
A073934(n) = if(1==n,n,n + A073934(n-(n/vecmin(factor(n)[,1]))));
isA333789(n) = (A073934(n)!=A333790(n));

A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

Original entry on oeis.org

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

Offset: 1

Ali Sada and Robert G. Wilson v, Feb 28 2020

From Antti Karttunen, Apr 07 2020: (Start)
Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).
Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.
The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.
For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.
(End)
A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is {12, 8}, taking 2 iterations to reach 8 = 2^3. So a(15) is 2.
From _Antti Karttunen_, Apr 07 2020: (Start)
Considering all possible paths from 15 to 1 nondeterministic map k -> k-(k/p), where p can be any prime factor of k, we obtain the following graph:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.
It can be seen that there's also alternative route to 8 via 10 (with 10 = 15-(15/3), where 3 is not the largest prime factor of 15), but it's not any shorter than the route via 12.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree labeling the index n, with a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value in a(n) for n = 1..16383.

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for positions of 0 .. 6 in this sequence, and also array A334100.
Cf. A334099 (a right inverse, positions of the first occurrence of each n).
Cf. A334091 (first differences), A335429 (partial sums).
Cf. also A331410 (analogous sequence when using the map k -> k + k/p), A334861, A335877 (their sums and differences), see also A335878 and A335884, A335885.

a[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] -1; Array[a, 100]

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 07 2020

up_to = 2^24;
A329697list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = if(!bitand(n,n-1),0,1+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v329697 = A329697list(up_to);
A329697(n) = v329697[n]; \\ Antti Karttunen, Apr 07 2020

A329697(n) = if(n<=2,0, if(isprime(n), A329697(n-1)+1, my(f=factor(n)); (apply(A329697, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, Apr 19 2020

From Antti Karttunen, Apr 07-19 2020: (Start)
a(1) = a(2) = 0; and for n > 2, a(p) = 1 + a(p-1) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [This is otherwise equal to the definition of A064097, except here we have a different initial condition, with a(2) = 0].
a(2n) = a(A000265(n)) = a(n).
a(p) = 1+a(p-1), for all odd primes p.
If A209229(n) == 1 [when n is a power of 2], a(n) = 0,
  otherwise a(n) = 1 + a(n-A052126(n)) = 1 + a(A171462(n)).
Equivalently, for non-powers of 2, a(n) = 1 + a(n-(n/A078701(n))),
or equivalently, for non-powers of 2, a(n) = 1 + Min a(n - n/p), for p prime and dividing n.
a(n) = A064097(n) - A064415(n), or equally, a(n) = A064097(n) - A054725(n), for n > 1.
a(A019434(n)) = 1, a(A334092(n)) = 2, a(A334093(n)) = 3, etc. for all applicable n.
For all n >= 0, a(A334099(n)) = a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = a(257^n) = a(65537^n) = n.
a(A122111(n)) = A334107(n), a(A225546(n)) = A334109(n).
(End)
From Antti Karttunen, Mar 16 2021: (Start)
a(n) = a(A336466(n)) + A087436(n) = A336396(n) + A087436(n).
a(A053575(n)) = A336469(n) = a(n) - A005087(n).
a(A147545(n)) = A000120(A147545(n)) - 1.
(End)

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!]

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).

A333001 The average path sum (floored down) 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, 23, 34, 25, 38, 37, 39, 31, 48, 41, 60, 46, 60, 63, 86, 50, 71, 71, 68, 71, 100, 74, 105, 63, 104, 89, 108, 81, 118, 112, 116, 90, 131, 112, 155, 119, 122, 153, 200, 101, 161, 132, 148, 135, 188, 131, 179, 137, 178, 181, 240, 144, 205, 192, 181, 127, 206, 191, 258, 170, 251, 199, 270, 160, 233, 218, 216

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}, with path sums 27, 25, 24, whose average is 76/3 = 25.333..., therefore a(12) = 25.
For n=15 we have five alternative paths from 15 to 1 (illustrated below) with path sums 37, 40, 42, 40, 39, whose average is 198/5 = 39.6, therefore a(15) = 39.
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \_   |  __/|
     \__|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.

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

Cf. A333000, A333123.
Cf. A333002/A333003 (average as exact rational, numerator/denominator in lowest terms), A333785 (where the average is an integer).
Cf. A333790 (smallest path sum), A333794 (conjectured largest path sum).

Map[Floor@ Mean[Total /@ #] &, #] &@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 74] (* Michael De Vlieger, Apr 15 2020 *)

up_to = 20000;
A333001list(up_to) = { my(u=vector(up_to), v=vector(up_to)); u[1] = v[1] = 1; for(n=2,up_to, my(ps=factor(n)[, 1]~); u[n] = vecsum(apply(p -> u[n-n/p], ps)); v[n] = (u[n]*n)+vecsum(apply(p -> v[n-n/p], ps))); vector(up_to, n, floor(v[n]/u[n])); };
v333001 = A333001list(up_to);
A333001(n) = v333001[n];

a(n) = floor(A333000(n)/A333123(n)) = floor(A333002(n)/A333003(n)).

cp's OEIS Frontend

A333789 Numbers k for which A333790(k) < A073934(k).

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A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

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A333794 a(1) = 1, for n > 1, a(n) = n + a(n-A052126(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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A333001 The average path sum (floored down) 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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