A098115 - OEIS

%I #13 Nov 19 2024 02:56:53

%S 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,

%T 56,17,36,40,123,57,61,9,99,94,132,158,172,23,43,89,186,196,194,203,

%U 157,205,62,32,26,76,105,65,45,177,56,278

%N 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).

%C 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.

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

%e 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.

%t 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}]

%t a[n_] := Length@ NestWhileList[(# - EulerPhi[#])&, Times @@ Prime[Range[2, n]], # > 0 &]; Array[a, 30] (* _Amiram Eldar_, Nov 19 2024 *)

%o (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

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

%K nonn,more,changed

%O 1,1

%A _Labos Elemer_, Sep 27 2004

%E a(38)-a(57) from _Amiram Eldar_, Nov 19 2024