A306762 Smallest integer k such that Sum_(i=1..k) lambda(i) is divisible by n, where lambda(i) is the Carmichael lambda function.

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, 17, 55, 10, 58, 15, 71, 16, 44, 19, 169, 24, 100, 11, 48, 12, 25, 49, 18, 26, 38, 30, 40, 23, 164, 28, 50, 13, 141, 20, 47, 55, 21, 14, 80, 58, 192, 15, 110, 71, 76

Offset: 1

Author

Michel Lagneau, Mar 08 2019

Keywords

• nonn
• easy

Examples

			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.
		

Crossrefs

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

Cf. A053049 (analog with totient function).

Programs

Maple

S:= ListTools:-PartialSums(map(numtheory:-lambda, [$1..500])):
N:= 100: count:= 0: V:= Vector(N):
for n from 1 to 500 while count < N do
   d:= select(t -> t <= N and V[t] = 0, numtheory:-divisors(S[n]));
   count:= count + nops(d);
   V[convert(d,list)]:= n;
od:
convert(V,list); # Robert Israel, Mar 11 2019

Mathematica

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

PARI

lambda(n) = lcm(znstar(n)[2]);
a(n) = {my(k=1, s=lambda(k)); while (s % n, k++; s += lambda(k)); k;} \\ Michel Marcus, Mar 09 2019