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Smallest integer k such that Sum_(i=1..k) lambda(i) is divisible by n, where lambda(i) is the Carmichael lambda function.
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%I #17 Mar 11 2019 20:15:55

%S 1,2,4,3,5,4,12,11,7,5,49,6,9,12,10,15,16,7,24,8,12,49,26,30,23,9,13,

%T 17,55,10,58,15,71,16,44,19,169,24,100,11,48,12,25,49,18,26,38,30,40,

%U 23,164,28,50,13,141,20,47,55,21,14,80,58,192,15,110,71,76

%N Smallest integer k such that Sum_(i=1..k) lambda(i) is divisible by n, where lambda(i) is the Carmichael lambda function.

%e a(7) = 12 because Sum_{i=1..12} lambda(i) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 2 + 6 + 4 + 10 + 2 = 42, and 42/7 = 6.

%p S:= ListTools:-PartialSums(map(numtheory:-lambda, [$1..500])):

%p N:= 100: count:= 0: V:= Vector(N):

%p for n from 1 to 500 while count < N do

%p d:= select(t -> t <= N and V[t] = 0, numtheory:-divisors(S[n]));

%p count:= count + nops(d);

%p V[convert(d,list)]:= n;

%p od:

%p convert(V,list); # _Robert Israel_, Mar 11 2019

%t a[n_] := (m = 1; While[! IntegerQ[Sum[CarmichaelLambda[k], {k, 1, m}]/n], m++]; m); a /@ Range[80]

%o (PARI) lambda(n) = lcm(znstar(n)[2]);

%o a(n) = {my(k=1, s=lambda(k)); while (s % n, k++; s += lambda(k)); k;} \\ _Michel Marcus_, Mar 09 2019

%Y Cf. A002322 (Carmichael lambda), A162578 (partial sums of A002322).

%Y Cf. A053049 (analog with totient function).

%K nonn,easy

%O 1,2

%A _Michel Lagneau_, Mar 08 2019