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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.)