A070179 -id:A070179 - cp's OEIS frontend

A070183 Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.

17, 41, 137, 401, 433, 449, 457, 521, 569, 641, 761, 809, 857, 919, 929, 953, 977, 1361, 1409, 1423, 1657, 1697, 1999, 2017, 2081, 2143, 2153, 2287, 2297, 2417, 2609, 2633, 2729, 2753, 2777, 2791, 2801, 2897, 2953, 3041, 3209, 3329, 3457, 3593, 3617

Offset: 1

Klaus Brockhaus, Apr 29 2002

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040992, A049568, A059264, A059667, A070179, A070180, A070181, A070182, A070184, A070185, A070186, A070187, A070188.

[p: p in PrimesUpTo(5000) | not exists{x: x in ResidueClassRing(p) | x^36 eq 2} and exists{x: x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 21 2012

select(p -> isprime(p) and [msolve(x^6=2,p)]<>[] and [msolve(x^36=2,p)]=[] , [seq(i,i=3..10^4,2)]); # Robert Israel, May 13 2018

PARI
```
forprime(p=2,3700,x=0; while(x
				
```

ok(p, r, k1, k2)={
    if (  Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
    if (  Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
    return(1);
}
forprime(p=2,10^4, if (ok(p,2,6,6^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

from itertools import count, islice
from sympy import nextprime, is_nthpow_residue
def A070183_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        if is_nthpow_residue(2,6,p) and not is_nthpow_residue(2,36,p):
            yield p
        p = nextprime(p)
A070183_list = list(islice(A070183_gen(),20)) # Chai Wah Wu, May 02 2024

A070187 Primes p such that x^11 = 2 has a solution mod p, but x^(11^2) = 2 has no solution mod p.

Original entry on oeis.org

12101, 13553, 30493, 32429, 44771, 66067, 103577, 128987, 180533, 182711, 187793, 201829, 242243, 257489, 264749, 299113, 314359, 330331, 337349, 341947, 356467, 371471, 431729, 442619, 475289, 484243, 505781, 513767, 540871, 558053, 564103, 573299, 581527, 582011, 586367, 593869, 596047, 630169

Offset: 1

Klaus Brockhaus, Apr 29 2002

nonn

Cf. A049543, A059667, A070179 - A070186, A070188.

PARI
```
forprime(p=2,550000,x=0; while(x
				
```

N=10^6;  default(primelimit,N);
ok(p, r, k1, k2)={
    if (  Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
    if (  Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
    return(1);
}
forprime(p=2,N, if (ok(p,2,11,11^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

A294912 Numbers n such that 2^(n-1), (2n-1)(2^((n-1)/2)), (4ceiling((3/4)n)-2), and (2^((n+1)/2) + floor((1/4)n)2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259

Offset: 1

Jonas Kaiser, Nov 10 2017

nonn

It appears that A007520 is a subsequence.
The first composite term is a(9969) = 476971 = 11*131*331. - Alois P. Heinz, Nov 10 2017
From Hilko Koning, Dec 03 2019: (Start)
The next composite terms < 1999979 are
a(17428) = 877099  = 307*2857
a(25090) = 1302451 = 571*2281
a(25518) = 1325843 = 499*2657
a(26785) = 1397419 = 67*20857
a(27549) = 1441091 = 347*4153
a(28715) = 1507963 = 971*1553
a(29117) = 1530787 = 619*2473
a(35635) = 1907851 = 11*251*691
(End)
From Hilko Koning, Dec 05 2019: (Start)
The next composite terms < 24999971 are
a(37344) = 2004403 = 307*6529
a(55773) = 3090091 = 1163*2657
a(56189) = 3116107 = 883*3529
a(91332) = 5256091 = 811*6481
a(102027) = 5919187 = 1777*3331
a(133230) = 7883731 = 811*9721
a(156407) = 9371251 = 1531*6121
a(182911) = 11081459 = 227*48817
a(189922) = 11541307 = 1699*6793
a(201043) = 12263131 = 811*15121
a(213203) = 13057787 = 467*27961
a(217484) = 13338371 = 3163*4217
a(257526) = 15976747 = 3739*4273
a(274961) = 17134043 = 1097*15619
a(299096) = 18740971 = 1531*12241
a(308928) = 19404139 = 2011*9649
a(321676) = 20261251 = 2251*9001
a(341902) = 21623659 = 1163*18593
a(348622) = 22075579 = 163*135433
a(380162) = 24214051 = 281*86171
The composite terms < 25*10^6 match the terms of A244628.
(End)
It appears that composites of the form 2k+1 such that 3*(2k+1) divides 2^k+1 are the composite terms of this sequence. - Hilko Koning, Dec 09 2019

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A244626, A294717, A293394, A070179, A007520.

okQ[n_] := AllTrue[{2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*Ceiling@((3/4)*n) - 2), (2^((n+1)/2) + Floor@(n/4)*2^(((n+1)/2)+1))}, Mod[#, n] == 1&];
Select[Range[1300], okQ] (* Jean-François Alcover, Feb 18 2019 *)

isok(n) = (n%2) && lift((Mod(2, n)^(n-1))==1)&&lift((Mod((2*n-1), n)*Mod(2, n)^((n-1)/2)) == 1)&&lift((Mod(((4*ceil((3/4)*n)-2)), n) )== 1)&&lift((Mod(2, n)^((n+1)/2) +Mod(floor((1/4)*n),n)*Mod(2, n)^(((n+1)/2)+1 ))== 1)

More terms from Alois P. Heinz, Nov 10 2017

A070182 Primes p such that x^5 = 2 has a solution mod p, but x^(5^2) = 2 has no solution mod p.

Original entry on oeis.org

151, 251, 3251, 3301, 4751, 8501, 11251, 11701, 13751, 14251, 14951, 15551, 16451, 17401, 18401, 21401, 21601, 24251, 28351, 28901, 32251, 32401, 32801, 34301, 36151, 36451, 37201, 40351, 42451, 42701, 44201, 45751, 46051, 46451, 46901

Offset: 1

Klaus Brockhaus, Apr 29 2002

Cf. A040159, A049557, A059313, A059667, A070179 - A070181, A070183 - A070188.

[p: p in PrimesUpTo(50000) | not exists{x: x in ResidueClassRing(p) | x^25 eq 2} and exists{x: x in ResidueClassRing(p) | x^5 eq 2}]; // Vincenzo Librandi, Sep 21 2012

PARI
```
forprime(p=2,47000,x=0; while(x
				
```

ok(p, r, k1, k2)={
    if (  Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
    if (  Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
    return(1);
}
forprime(p=2,10^5, if (ok(p,2,5,5^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

A294919 Numbers n such that 2^(n-1), (2n-1)(2^((n-1)/2)), (4ceiling((1/4)n)-2), and (2^((n+1)/2) + floor((3/4)n)2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

Original entry on oeis.org

5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 541, 557, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 877, 941, 997, 1013, 1021, 1061, 1069, 1093, 1109, 1117, 1181, 1213

Offset: 1

Jonas Kaiser, Nov 10 2017

nonn

It appears that A007521 is a subsequence.

a(118) = 3277 = 29*113 is the first nonprime term.

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A007521, A070179, A244626, A293394, A294717.

okQ[n_] := AllTrue[{2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*Ceiling@(n/4) - 2), (2^((n+1)/2) + Floor@((3/4)*n)*2^(((n+1)/2) + 1))}, Mod[#, n] == 1&];
Select[Range[1300], okQ] (* Jean-François Alcover, Feb 18 2019 *)

isok(n) = (n%2) && lift((Mod(2, n)^(n-1))==1)&&lift((Mod((2*n-1), n)*Mod(2, n)^((n-1)/2)) == 1)&&lift((Mod(((4*ceil((1/4)*n)-2)), n) )== 1)&&lift((Mod(2, n)^((n+1)/2) +Mod(floor((3/4)*n),n)*Mod(2, n)^(((n+1)/2)+1 ))== 1)

More terms from Alois P. Heinz, Nov 10 2017

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2n-1)(2^((n-1)/2)), 4ceiling((3/4)n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).

Original entry on oeis.org

11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283

Offset: 1

Jonas Kaiser, Nov 12 2017

nonn

It appears that A007520 is a subsequence. Up to 10^7 there are no composites in this sequence.

The first composite is a(17465859) = 1397357851; there are probably infinitely many. - Charles R Greathouse IV, Nov 12 2017

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A244626, A294717, A293394, A070179, A007520.

Select[Range[2, 1300], Function[n, AllTrue[Join[Prime[Range@3]^(n - 1), {(2 n - 1) (2^((n - 1)/2)), 4 Ceiling[3 n/4] - 2, (2^((n + 1)/2) + Floor[n/4]*2^((n + 3)/2))}], Mod[#, n] == 1 &]]] (* Michael De Vlieger, Nov 15 2017 *)

is(n) = n%2 && Mod(2, n)^(n-1)==1 && Mod(3, n)^(n-1)==1 && Mod(5, n)^(n-1)==1 && (2*n-1)*Mod(2, n)^((n-1)/2)== 1 && Mod(4*ceil((3/4)*n)-2, n)==1 && Mod(2, n)^((n+1)/2)+floor(n/4)*Mod(2, n)^((n+3)/2)==1

cp's OEIS Frontend

A070183 Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.

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A070187 Primes p such that x^11 = 2 has a solution mod p, but x^(11^2) = 2 has no solution mod p.

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A294912 Numbers n such that 2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*ceiling((3/4)*n)-2), and (2^((n+1)/2) + floor((1/4)*n)*2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

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A070182 Primes p such that x^5 = 2 has a solution mod p, but x^(5^2) = 2 has no solution mod p.

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Author

Keywords

Crossrefs

Programs

Magma

PARI

PARI

A294919 Numbers n such that 2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*ceiling((1/4)*n)-2), and (2^((n+1)/2) + floor((3/4)*n)*2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

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Programs

Mathematica

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Extensions

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2*n-1)*(2^((n-1)/2)), 4*ceiling((3/4)*n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A294912 Numbers n such that 2^(n-1), (2n-1)(2^((n-1)/2)), (4ceiling((3/4)n)-2), and (2^((n+1)/2) + floor((1/4)n)2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

A294919 Numbers n such that 2^(n-1), (2n-1)(2^((n-1)/2)), (4ceiling((1/4)n)-2), and (2^((n+1)/2) + floor((3/4)n)2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2n-1)(2^((n-1)/2)), 4ceiling((3/4)n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).