A098115 a(n) is the length of iteration trajectory when the cototient function (A051953) is applied to the half of the n-th primorial number (A070826(n) = A002110(n)/2).

2, 3, 4, 7, 10, 5, 12, 15, 12, 28, 6, 6, 31, 12, 47, 29, 23, 32, 33, 24, 40, 28, 12, 35, 34, 56, 17, 36, 40, 123, 57, 61, 9, 99, 94, 132, 158, 172, 23, 43, 89, 186, 196, 194, 203, 157, 205, 62, 32, 26, 76, 105, 65, 45, 177, 56, 278

Offset

1,1

Comments

Initial values are here odd numbers. Comparing with the case of primorials (A098202), the lengths are here significantly smaller. The cause of this is unknown, albeit informally "understood": lack of powers of 2 in lists because parity is invariant during this iteration. See also lists for A098200 and A098201.

Links

Table of n, a(n) for n=1..57.

Formula

a(n) = A053475(A070826(n)) = A053475(A002110(n)/2).

Example

For n = 7: list = {255255,163095,77815,16663,895,183,63,27,9,3,1,0}, a(7) = 12, while the comparable length for 510510 is A098202(7) = 43.

Mathematica

g[x_] :=x-EulerPhi[x]; f[x_] :=Length[FixedPointList[g, x]]-1; q[x_] :=Apply[Times, Table[Prime[j], {j, 1, x}]]; t=Table[f[q[w]/2], {w, 1, 37}]
a[n_] := Length@ NestWhileList[(# - EulerPhi[#])&, Times @@ Prime[Range[2, n]], # > 0 &]; Array[a, 30] (* Amiram Eldar, Nov 19 2024 *)

Prog

(PARI) a(n) = {my(p = prod(i=2, n, prime(i)), c = 1); while(p > 0, c++; p -= eulerphi(p)); c; } \\ Amiram Eldar, Nov 19 2024

Crossrefs

Cf. A002110, A051953, A053475, A070826, A098200, A098201, A098202.

Sequence in context: A256772 A274976 A269729 * A182833 A182832 A375723

Adjacent sequences: A098112 A098113 A098114 * A098116 A098117 A098118

Keyword

nonn,more,changed

Author

Labos Elemer, Sep 27 2004

Extensions

a(38)-a(57) from Amiram Eldar, Nov 19 2024

Status

approved