A333003 -id:A333003 - cp's OEIS frontend

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.

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)).

A333785 Numbers m such that the average path sum is an integer when iterating from m to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 14, 16, 17, 20, 25, 32, 50, 64, 68, 82, 83, 84, 100, 101, 125, 128, 162, 163, 170, 235, 243, 256, 257, 272, 289, 456, 457, 512, 548, 621, 1024, 1028, 1040, 2048, 4096, 4112, 7225, 8192, 8738, 9248, 13058, 16384, 16480, 16481, 17476, 17477, 32128, 32768, 65536, 65537, 131072, 132098, 262144, 262148, 264196

Offset: 1

Antti Karttunen, Apr 06 2020

nonn

Numbers m such that A333123(m) divides A333000(m).

Positions of ones in A333003.

Antti Karttunen, Table of n, a(n) for n = 1..75; all terms up to <= 2^23

Cf. A333000, A333001, A333003, A333123.

Subsequences: A000079, A019434.

Position[Map[Mean[Total /@ #] &, #], ?IntegerQ][[All, 1]] &@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 2^10 + 1] (* _Michael De Vlieger, Apr 15 2020 *)

A333002 Numerator of the average 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, 25, 39, 15, 43, 47, 69, 76, 115, 37, 198, 31, 48, 209, 304, 46, 302, 317, 432, 203, 71, 500, 344, 640, 901, 899, 1271, 63, 1777, 179, 758, 736, 1069, 786, 465, 361, 525, 789, 1090, 358, 860, 1075, 1404, 506, 1132, 132, 1042, 815, 1133, 918, 1439, 965, 1251, 4165, 5522, 3026, 4307, 6343, 1273, 127

Offset: 1

Antti Karttunen, Apr 06 2020

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

See A333003 for denominator.

Cf. A333000, A333001, A333123.

Map[Numerator@ 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}}}, 63] (* Michael De Vlieger, Apr 15 2020 *)

up_to = 20000;
A333002list(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, numerator(v[n]/u[n])); };
v333002 = A333002list(up_to);
A333002(n) = v333002[n];

a(n) = numerator(A333000(n)/A333123(n)).

cp's OEIS Frontend

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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A333785 Numbers m such that the average path sum is an integer when iterating from m to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

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A333002 Numerator of the average 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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