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A303643 Numbers k such that k and phi(k) are in A292544.

%I A303643 #43 May 14 2018 10:18:49
%S A303643 1,1664,6815744,10092544,27917287424,4707284156416,5506148073472,
%T A303643 7060926234624,8259222110208,114349209288704,108649341010313216,
%U A303643 468374361246531584,1918461383665793368064,7858017827495089635590144,11635911013790805806546944,183907840308875463202177024
%N A303643 Numbers k such that k and phi(k) are in A292544.
%F A303643 Following steps can be used in order to produce terms of this sequence.
%F A303643 (1) Take odd m, find z and k (see formula section of A292544).
%F A303643 (2) Represent phi(m) = 2^t*m', where m' is odd (i.e., m' = A053575(m)).
%F A303643 (3) For this m', find z' and k'.
%F A303643 (4) Solve z*i - k + t = z'*j - k' + 1 for positive i, j.
%F A303643 (5) Each such solution gives a term m*2^(z*i - k + 1) of this sequence.
%F A303643 For all x >= 0, 13*2^(12*x+7), 77*2^(60*x+17), 137*2^(136*x+35), 173*2^(1204*x+259), 193*2^(96*x+49), 269*2^(8844*x+6567), 411*2^(136*x+34), 519*2^(1204*x+258), 557*2^(38364*x+28635), 563*2^(19670*x+9836), 581*2^(2460*x+789), 641*2^(64*x+33), 653*2^(52812*x+39447), 667*2^(4620*x+3405), 769*2^(384*x+193), 807*2^(8844*x+6566) are terms of this sequence (m < 10^3 where m*2^(z*i - k + 1) is the corresponding form).
%e A303643 1 is in A292544, and eulerphi(1)=1, so 1 is a term.
%e A303643 1664 and 768=eulerphi(1668) are both in A292544, so 1664 is a term.
%o A303643 (PARI) isA292544(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);
%o A303643 isok(n) = isA292544(n) && isA292544(eulerphi(n));
%o A303643 (PARI) { ZK(m) = my(z,k); z=znorder(Mod(2,m)); k=znlog(eulerphi(m),Mod(2,m)); if(type(k)!="t_INT",return()); [z,k]; }
%o A303643 { getpowerof2(m) = my(m2,t,zk,zk2,r);
%o A303643     m2 = eulerphi(m);
%o A303643     t = valuation(m2,2);
%o A303643     m2 \= 2^t;
%o A303643     if( m2==1, return(0));
%o A303643     zk=ZK(m);
%o A303643     zk2=ZK(m2);
%o A303643     if(!zk || !zk2, return());
%o A303643     r = [zk[1],zk2[1],zk[2]-t-zk2[2]+1];  \\ solving r[1] * i = r[2] * j + r[3]
%o A303643     r /= content(r);
%o A303643     if( gcd(r[1],r[2])>1, return());
%o A303643     ((r[2]*lift(Mod(-r[3]/r[2],r[1])) + r[3])/r[1] + r[2]*x)*zk[1] - zk[2] + 1;} \\ getpowerof2(m) returns z*i - k + 1 with x parameter (see formula section), i.e., getpowerof2(13) returns 12*x+7, that is, 13*2^(12*x+7) is a term for all x >= 0.
%Y A303643 Cf. A007733, A053575, A066781, A292544.
%K A303643 nonn
%O A303643 1,2
%A A303643 _Max Alekseyev_ and _Altug Alkan_, Apr 27 2018