A333794 -id:A333794 - cp's OEIS frontend

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

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.

A378524 Dirichlet inverse of A333794.

Original entry on oeis.org

1, -3, -6, 2, -12, 23, -20, 0, 14, 47, -36, -27, -40, 79, 102, 0, -48, -81, -64, -57, 174, 143, -96, 10, 68, 159, -24, -97, -112, -517, -116, 0, 314, 191, 360, 170, -128, 255, 350, 22, -144, -885, -176, -177, -400, 383, -240, 0, 218, -393, 414, -197, -216, 211, 656, 38, 566, 447, -284, 947, -232, 463, -696, 0, 724

Offset: 1

Antti Karttunen, Dec 01 2024

sign

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

Cf. A052126, A333794.

Cf. also A378523.

A052126(n) = if(1==n,n,(n/vecmax(factor(n)[, 1])));
A333794(n) = if(1==n,n,n + A333794(n-A052126(n)));
memoA378524 = Map();
A378524(n) = if(1==n,1,my(v); if(mapisdefined(memoA378524,n,&v), v, v = -sumdiv(n,d,if(dA333794(n/d)*A378524(d),0)); mapput(memoA378524,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA333794(n/d) * a(d).

A332993 a(1) = 1, for n > 1, a(n) = n + a(A032742(n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 10, 8, 15, 13, 16, 12, 22, 14, 22, 21, 31, 18, 31, 20, 36, 29, 34, 24, 46, 31, 40, 40, 50, 30, 51, 32, 63, 45, 52, 43, 67, 38, 58, 53, 76, 42, 71, 44, 78, 66, 70, 48, 94, 57, 81, 69, 92, 54, 94, 67, 106, 77, 88, 60, 111, 62, 94, 92, 127, 79, 111, 68, 120, 93, 113, 72, 139, 74, 112, 106, 134, 89, 131, 80, 156, 121

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

Sum of those divisors of n that can be obtained by repeatedly taking the largest proper divisor (of previous such divisor, starting from n, which is included in the sum), up to and including the terminal 1.

			a(18) = 18 + 18/2 + 9/3 + 3/3 = 18 + 9 + 3 + 1 = 31.

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, A006022, A032742, A064097, A073934, A332994, A333783, A333791, A333794.

A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));

a(1) = 1; and for n > 1, a(n) = n + a(A032742(n)).
a(n) = n + A006022(n).
a(n) = A332994(n) + A333791(n).
a(n) = A000203(n) - A333783(n).
It seems that for all n >= 1, a(n) <= A073934(n) <= A333794(n).

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).

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)).

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).

A333792 a(1) = 0, then after the first differences of A333793.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, -1, 1, 2, 0, 0, 0, 5, -3, -5, 0, 6, 0, 3, 3, 6, 0, -11, 2, 9, -12, 16, 0, -4, 0, -18, 30, -15, 9, -8, 0, 17, -5, -7, 0, 16, 0, 11, -24, 45, 0, -54, 22, -1, -7, 19, 0, -23, 29, -4, -1, 28, 0, -33, 0, 29, -24, -49, 56, 27, 0, -38, 68, -38, 0, -39, 0, 35, -17, 33, 0, -6, 0, -36, -20, 59, 0, 3, -30, 71, -9, -11, 0, -37, 16

Offset: 1

Antti Karttunen, Apr 05 2020

sign

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

Cf. A073934, A333793, A333794.

A333792(n) = if(1==n,0,A333793(n)-A333793(n-1)); \\ Other code as in A333793.

a(1) = 0; and for n > 1, a(n) = A333793(n) - A333793(n-1).

a(p) = 0 for all primes p. (May obtain zero values also on some nonprimes).

cp's OEIS Frontend

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

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A378524 Dirichlet inverse of A333794.

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A332993 a(1) = 1, for n > 1, a(n) = n + a(A032742(n)).

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A332994 a(1) = 1, for n > 1, a(n) = n + a(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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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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Mathematica

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A333792 a(1) = 0, then after the first differences of A333793.

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