A292544 -id:A292544 - cp's OEIS frontend

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

1, 1664, 6815744, 10092544, 27917287424, 4707284156416, 5506148073472, 7060926234624, 8259222110208, 114349209288704, 108649341010313216, 468374361246531584, 1918461383665793368064, 7858017827495089635590144, 11635911013790805806546944, 183907840308875463202177024

Offset: 1

Max Alekseyev and Altug Alkan, Apr 27 2018

nonn

			1 is in A292544, and eulerphi(1)=1, so 1 is a term.
1664 and 768=eulerphi(1668) are both in A292544, so 1664 is a term.

Cf. A007733, A053575, A066781, A292544.

isA292544(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);
isok(n) = isA292544(n) && isA292544(eulerphi(n));

{ 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]; }
{ getpowerof2(m) = my(m2,t,zk,zk2,r);
    m2 = eulerphi(m);
    t = valuation(m2,2);
    m2 \= 2^t;
    if( m2==1, return(0));
    zk=ZK(m);
    zk2=ZK(m2);
    if(!zk || !zk2, return());
    r = [zk[1],zk2[1],zk[2]-t-zk2[2]+1];  \\ solving r[1] * i = r[2] * j + r[3]
    r /= content(r);
    if( gcd(r[1],r[2])>1, return());
    ((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.

Following steps can be used in order to produce terms of this sequence.
(1) Take odd m, find z and k (see formula section of A292544).
(2) Represent phi(m) = 2^t*m', where m' is odd (i.e., m' = A053575(m)).
(3) For this m', find z' and k'.
(4) Solve z*i - k + t = z'*j - k' + 1 for positive i, j.
(5) Each such solution gives a term m*2^(z*i - k + 1) of this sequence.
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).

A318262 Numbers m such that 2^phi(m) mod m is a prime power (in the sense of A246655).

Original entry on oeis.org

6, 12, 14, 20, 24, 28, 30, 40, 48, 56, 60, 62, 72, 80, 84, 96, 112, 120, 124, 126, 144, 168, 192, 224, 240, 248, 252, 254, 272, 288, 320, 336, 340, 384, 408, 448, 480, 496, 504, 508, 510, 544, 576, 584, 640, 672, 680, 768, 816, 896, 960, 992, 1008, 1016, 1020

Offset: 1

Peter Luschny, Sep 03 2018

nonn

m is in this sequence if and only if 2^phi(m) mod m = 2^k for some k > 0.

There is no prime power in this sequence. Perfect power terms of this sequence are 144, 576, 9216, 36864, 589824, 884736, 1638400, 2359296, 3211264, 6553600, 7077888, ... - Altug Alkan, Sep 04 2018

			The odd part of the first few terms can be arranged as follows:
3,
3, 7,                         5,
3, 7, 15,                     5,
3, 7, 15, 31,              9, 5, 21,
3, 7, 15, 31, 63,          9,    21,
3, 7, 15, 31, 63, 127, 17, 9, 5, 21, 85,

Cf. A000010, A001597, A118372, A246655, A292544, A318623, A318145.

Select[Range[2^10], And[PrimePowerQ@ #, ! PrimeQ@ #] &@ Mod[2^EulerPhi@ #, #] &] (* Michael De Vlieger, Sep 04 2018 *)

isok(n) = isprimepower(lift(Mod(2, n)^eulerphi(n))); \\ Michel Marcus, Sep 06 2018

def isA318262(n):
    m = power_mod(2, euler_phi(n), n)
    return m.is_prime_power()
def A318262_list(search_bound):
    return [n for n in range(2,search_bound+1,2) if isA318262(n)]
print(A318262_list(1020))

A292742 Odd numbers k such that 2^psi(k) == phi(k) (mod k).

Original entry on oeis.org

1, 5, 10393, 45411263, 31761301667, 70697569679

Offset: 1

Altug Alkan, Sep 22 2017

a(7) > 5*10^11. - Giovanni Resta, Sep 23 2017

			10393 = 19*547 is a term because 2^((19 + 1)*(547 + 1)) == (19 - 1)*(547 - 1) (mod 19*547).

Cf. A000010, A001615, A292544.

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = n%2==1 && Mod(2, n)^a001615(n)==eulerphi(n); \\ after Charles R Greathouse IV at A001615

a(5)-a(6) from Giovanni Resta, Sep 23 2017

A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 10, 12, 384, 640, 768, 896, 960, 24576, 49152, 950272, 1425408, 1572864, 3145728, 10485760, 19398656, 65011712, 100663296, 110057537, 201326592, 220115074, 671088640, 1879048192

Offset: 1

Altug Alkan, Sep 17 2018

Sequence is infinite, i.e., 3*2^(3*(t-1)-(-1)^t) is a term for all t > 0.

Prime terms (5, 110057537, ...) are in A246568 based on case A = +1.

Cf. A077815, A077816, A246568, A292544.

[1] cat [n: n in [1..10^6] | 2^EulerPhi(n) mod n^2 eq EulerPhi(n)^2]; // Vincenzo Librandi, Sep 20 2018

isok(n) = Mod(2, n^2)^eulerphi(n)==eulerphi(n)^2;

cp's OEIS Frontend

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

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A318262 Numbers m such that 2^phi(m) mod m is a prime power (in the sense of A246655).

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Sage

A292742 Odd numbers k such that 2^psi(k) == phi(k) (mod k).

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A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

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