A248881 -id:A248881 - cp's OEIS frontend

A257484 Numbers k such that lambda(sum of divisors of k that are congruent to 0 mod 3) = lambda(sum of divisors of k that are congruent to 1 mod 3) = lambda(sum of divisors of k that are congruent to 2 mod 3) where lambda is the Carmichael function (A002322).

78, 222, 234, 294, 312, 366, 375, 438, 582, 618, 666, 834, 876, 882, 888, 936, 942, 1086, 1095, 1098, 1125, 1158, 1176, 1236, 1314, 1464, 1482, 1536, 1545, 1662, 1746, 1752, 1842, 1878, 2013, 2022, 2028, 2094, 2166, 2274, 2316, 2328, 2382, 2472, 2502, 2526

Offset: 1

Michel Lagneau, Apr 26 2015

nonn

A majority of numbers having three distinct prime divisors are in the sequence, but the number 1482 contains four distinct prime divisors {2,3,13,19}.
If a(n) is a squarefree number (subsequence 78, 222, 366, 438, 582, 618, 834, 942, 1086, 1095, 1158, 1482, 1545, 1662, 1842, 1878, 2013, 2022, 2094, 2274, 2382, 2526,...), the number 3*a(n) is also in the sequence because the sum of divisors of a(n) that are congruent to 0 mod 3 is equal to k, the sum of divisors of 3*a(n) that are congruent to 0 mod 3 is equal to 4k, and lambda(4*k) = lcm(lambda(4),lambda(k)) = lcm(2,lambda(k)) = lambda(k).
The sequence of the corresponding values lambda is  {6, 18, 6, 18, 12, 30, 12, 36, 42, 12, 18, 12, 36, 18, 36, 12, 78, 12, 36, 30, 12, 96, 36, 12, 36, 60, 12, 30, 12, 138, 42, 36, 30, 156, 30, 156, 60, 60, 126, 36, 96, 84, 198, 12, 12, 210, 30,...}.

			78 is in the sequence because the divisors of 78 are {1,2,3,6,13,26,39,78} and the divisors congruent to 0 mod 3 are {3,6,39,78} => sum=126, the divisors congruent to 1 mod 3 are {1,13} => sum=14, the divisors congruent to 2 mod 3 are {2,26} => sum=28, and lambda(126)=lambda(14)=lambda(28) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002322, A248881.

with(numtheory):nn:=2600:
  for n from 1 to nn do:
   s0:=0:s1:=0:s2:=0:
   x:=divisors(n):n0:=nops(x):
    for i from 1 to n0 do:
     q:=x[i]:
      if irem(q,3)=0 then s0:=s0+q:
       else
        if irem(q,3)=1 then s1:=s1+q:
       else
        s2:=s2+q:
        fi:fi:
       od:
      if lambda(s0)=lambda (s1) and lambda(s1)=lambda(s2)
      then
      printf(`%d, `,n):
      else
      fi:
      od:

lst={}; f[x_] := Plus @@ Select[Divisors[x], Mod[#,3]==0 &]; g[x_] := Plus @@ Select[Divisors[x], Mod[#,3]==1 &];h[x_] := Plus @@ Select[Divisors[x], Mod[#,3]==2 &]; Do[If[CarmichaelLambda[f[n]]== CarmichaelLambda[g[n]]&& CarmichaelLambda[f[n]]== CarmichaelLambda[h[n]], AppendTo[lst, n]], {n, 1, 2600}]; lst

lambda(n)=lcm(znstar(n)[2]);
isok(n) = {my(sd0=sumdiv(n, d, d*((d % 3)==0))); my(sd1=sumdiv(n, d, d*((d % 3)==1))); my(sd2=sumdiv(n, d, d*((d % 3)==2))); sd0 && sd1 && sd2 && (lambda(sd0) == lambda(sd1)) && (lambda(sd0)==lambda(sd2));}
lista(nn) = for (n=1, nn, if (isok(n), print1(n, ", "))); \\ Michel Marcus, May 02 2015

cp's OEIS Frontend

A257484 Numbers k such that lambda(sum of divisors of k that are congruent to 0 mod 3) = lambda(sum of divisors of k that are congruent to 1 mod 3) = lambda(sum of divisors of k that are congruent to 2 mod 3) where lambda is the Carmichael function (A002322).

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