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COMMENT
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A068563[n] are the numbers n such that 2^n (mod n) = 4^n (mod n). A124240[n] are the numbers n such that A124239(n) is divisible by n. A124239[n] = Sum[ Sum[ (2k-1)^m, {m,1,n} ], {k,1,n} ] = n + Sum[ (2k-1)((2k-1)^n-1) / (2(k-1)), {k,2,n} ]. It appears that A124240[n] almost coincides with A068563[n]. The first term that is different is A068563[27] = 136. The second term that is dofferent is A068563[54] = 408. Note that a(2) = 3*a(1) and a(4) = 5*a(1).
a(6) = 1224 = 9*a(1), a(7) = 1240 = 2*a(3), a(8) = 1314, a(9) = 2040 = 15*a(1), a(10) = 2312 = 17*a(1), a(11) = 2460 = 3*a(5), a(12)= 2480 = 4*a(3), a(13) = 2856 = 21*a(1). Numbers k such that there exists a(n) = k*a(1) are k = {1, 3, 5, 9, 15, 17, 21, ...}. Most listed a(n) except two (a(7) = 1240 and a(12) = 2480) belong to A124276 = {1,6,18,20,42,54,60,100,126,136,156,162,180,220,294,300,342,378,408,...} Terms of A068563[n] such that A068563[n]/2 is not a term of A068563[n].
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