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

1,6

Links

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

Formula

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

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 *)

Prog

(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];

Crossrefs

See A333002 for numerator.

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

Sequence in context: A000003 A234398 A168208 * A352072 A365067 A332086

Adjacent sequences: A333000 A333001 A333002 * A333004 A333005 A333006

Keyword

nonn,frac

Author

Antti Karttunen, Apr 06 2020

Status

approved