A219793 Least k such that phi(n) = lambda(k), or 0 if there is no such k.

1, 1, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 13, 7, 32, 32, 17, 7, 19, 32, 13, 11, 23, 32, 25, 13, 19, 13, 29, 32, 31, 17, 25, 17, 224, 13, 37, 19, 224, 17, 41, 13, 43, 25, 224, 23, 47, 17, 43, 25, 128, 224, 53, 19, 41, 224, 37, 29, 59, 17, 61, 31, 37, 128, 119, 25

Offset: 1

Author

Michel Lagneau, Nov 28 2012

Keywords

• nonn

Comments

lambda(n) is the Carmichael lambda function A002322. For n <10000, it appears that a(n) = 0 for n = 2047, 4094, 6141, 6533, 8119, 8188, 9637. if a(n) = p is a prime greater than 2, then n belongs to the finite set {p, p1, p2, ...., pk} that is a subsequence of A032447 (see the array with characteristic rows in the example of A032447), for example : a(n) = 3 for n = 3, 4, 6; a(n) = 5 for n = 5, 8, 10, 12; a(n) = 7 for n = 7, 9, 14, 18, 15, 16, 20, 24, 30; a(n) = 11 for n = 11, 22; a(n) = 13 for n = 13, 21, 26, 28, 36, 42; a(n) = 17 for n = 17, 32, 34, 40, 48, 60.

Examples

			a(6) = 3 because phi(6) = lambda(3) = 2.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002322, A032447, A141162, A143407.

Programs

Maple

with(numtheory): for n from 1 to 100 do: ii:=0:for k from 1 to 10^6 while(ii=0) do:if phi(n)=lambda(k) then ii:=1: printf(`%d, `,k):else fi:od:if ii=0 then printf(`%d, `,0): else fi:od:

Mathematica

Table[k=0; While[!EulerPhi[n] == CarmichaelLambda[k], k++]; k, {n, 100}] (* program will go into an infinite loop at n = 2047 *)