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 A292544 Numbers h such that 2^phi(h) == phi(h) (mod h). 5
 1, 12, 40, 48, 60, 192, 544, 640, 680, 704, 768, 816, 960, 1020, 1664, 3072, 10240, 11008, 12288, 13760, 15360, 19456, 24320, 49152, 83968, 125952, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 720896, 786432 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: For n > 1, a(n) is a Zumkeller number (A083207) [confirmed for n up to 47]. - Ivan N. Ianakiev, Sep 22 2017 LINKS Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12; first 101 terms from Michel Marcus) FORMULA Let m be an odd number, z = A007733(m) and k, 0 <= k < z, be such that phi(m) == 2^k (mod m); then m*2^(i*z - k + 1) belongs to this sequence for all i >= 1. And this is a general form of the terms of this sequence. Some families of solutions of the form m*2^(i*z - k + 1): If m = 3, then z = 2 and k = 1 ==> 3*2^(2*i) is a term for all i >= 1. If m = 5, then z = 4 and k = 2 ==> 5*2^(4*i-1) is a term for all i >= 1. If m = 7, then z = 3 but k does not exist ==> no term with odd part equal to 7. If m = 15, then z = 4 and k = 3 ==> 15*2^(4*i-2) is a term for all i >= 1. If m = 77, then z = 30 and k = 14 ==> 77*2^(30*i-13) is a term for all i >= 1. EXAMPLE 704 = 11*2^6 is a term since phi(11*2^6) = 5*2^6 and 11*2^6 divides 2^(5*2^6) - 5*2^6. MATHEMATICA {1}~Join~Select[Range[10^6], Function[n, # == PowerMod[2, #, n] &@ EulerPhi@ n]] (* Michael De Vlieger, Sep 18 2017 *) PROG (PARI) isok(n) = Mod(2, n)^eulerphi(n)==eulerphi(n); CROSSREFS Cf. A000010, A007733, A066781. Sequence in context: A226348 A359023 A139691 * A345924 A114815 A363121 Adjacent sequences: A292541 A292542 A292543 * A292545 A292546 A292547 KEYWORD nonn,easy AUTHOR Max Alekseyev and Altug Alkan, Sep 18 2017 STATUS approved

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Last modified September 14 03:52 EDT 2024. Contains 375911 sequences. (Running on oeis4.)