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A257484 Numbers n such that lambda(sum of divisors of n that are congruent to 0 mod 3) = lambda(sum of divisors of n that are congruent to 1 mod 3) = lambda(sum of divisors of n that are congruent to 2 mod 3) where lambda is the Carmichael function (A002322). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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,...}.

LINKS

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

EXAMPLE

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.

MAPLE

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:

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

Cf. A002322, A248881.

Sequence in context: A044410 A044791 A232386 * A068130 A118938 A206004

Adjacent sequences:  A257481 A257482 A257483 * A257485 A257486 A257487

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 26 2015

STATUS

approved

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Last modified August 14 18:13 EDT 2022. Contains 356122 sequences. (Running on oeis4.)