A333003 Denominator 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.

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 1, 5, 1, 1, 5, 5, 1, 5, 5, 5, 4, 1, 7, 5, 9, 9, 12, 12, 1, 17, 2, 7, 9, 9, 7, 4, 4, 4, 7, 7, 3, 7, 7, 7, 5, 7, 1, 7, 6, 6, 7, 8, 7, 7, 23, 23, 21, 21, 33, 7, 1, 11, 47, 47, 1, 61, 28, 28, 7, 7, 23, 14, 2, 103, 3, 3, 5, 7, 1, 1, 1, 4, 21, 79, 7, 7, 7, 7, 7, 89, 7, 14, 2, 2, 21, 103, 1, 1, 16, 16, 18, 84

Offset: 1

Author

Antti Karttunen, Apr 06 2020

Keywords

• nonn
• frac

Links

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

Crossrefs

See A333002 for numerator.

Cf. A333000, A333001, A333123, A333785 (positions of ones).

Programs

Mathematica

Map[Denominator@ 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}}}, 104] (* Michael De Vlieger, Apr 15 2020 *)

PARI

up_to = 20000;
A333003list(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, denominator(v[n]/u[n])); };
v333003 = A333003list(up_to);
A333003(n) = v333003[n];

Formula

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