A015950 -id:A015950 - cp's OEIS frontend

A015969 Numbers k that divide 16^k + 1.

1, 17, 289, 4913, 83521, 1419857, 6029713, 12027313, 24137569, 85525793, 102505121, 204464321, 410338673, 1453938481, 1742587057, 3475893457, 6975757441, 24716954177, 29623979969, 59090188769, 111612202577, 118587876497, 420188221009, 500540685121, 503607659473

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..34 (terms < 10^13)

Cf. A006521, A015949, A015950, A015951, A015953, A015954, A015955, A015957, A015958, A015960, A015961, A015963, A015965, A015968, A014957.

Column k=16 of A333429.

More terms from Max Alekseyev, Oct 02 2010

Missing terms a(10), a(14), a(18), and a(23) from Giovanni Resta, Mar 23 2020

A327840 Numbers m that divide 4^m + 3.

Original entry on oeis.org

1, 7, 16387, 4509253, 24265177, 42673920001, 103949349763, 12939780075073

Offset: 1

Juri-Stepan Gerasimov, Sep 27 2019

Number of solutions < 10^9 to k^n == k-1 (mod n): 1 (if k = 1), 188 (if k = 2, see A006521), 5 (if k = 3, see A015973), 5 (if k = 4, see this sequence), 5 (if k = 5), 10 (if k = 6), 10 (if k = 7), 7 (if k = 8), 5 (if k = 9), 8 (if k = 10), 11 (if k = 11), 8 (if k = 12), 9 (if k = 13), 4 (if k = 14), 3 (if k = 15), 6 (if k = 16), 7 (if k = 17), 7 (if k = 18), ...

a(9) > 10^15. - Max Alekseyev, Nov 10 2022

Solutions to k^n == 1-k (mod n): A006521 (k = 2), A015973 (k = 3), this sequence (k = 4), A123047 (k = 5), A327943 (k = 6).
Solutions to 4^n == k (mod n): A000079 (k = 0), A015950 (k = -1), A014945 (k = 1), A130421 (k = 2), this sequence (k = -3), A130422 (k = 3).
Cf. A015940, A253208.

[1] cat [n: n in [1..10^8] | Modexp(4,n,n) + 3 eq n];

Select[Range[10^7], IntegerQ[(PowerMod[4, #, # ]+3)/# ]&] (* Metin Sariyar, Sep 28 2019 *)

is(n)=Mod(4,n)^n==-3 \\ Charles R Greathouse IV, Sep 29 2019

a(6)-a(7) from Giovanni Resta, Sep 29 2019

a(8) from Max Alekseyev, Nov 10 2022

A211349 Primes p such that p-1 divides 2^p + 2.

Original entry on oeis.org

2, 3, 11, 251, 5051, 16811, 2025251, 8751251, 16607051, 28257611, 69005051, 78906251, 176775251, 210381251, 372175451, 550427051, 707025251, 854704451, 1866788051, 2441406251, 2605806251, 4249701251, 5469531251, 9304386251, 10315761251, 10915095251

Offset: 1

Philip A. Hoskins, Feb 06 2013

nonn

Prime p>2 is in this sequence iff (p-1)/2 is in A015950. - Max Alekseyev, Dec 26 2017

Max Alekseyev, Table of n, a(n) for n = 1..177

Cf. A069051, A211203.

Select[Prime[Range[1000]], Mod[2^# + 2, # - 1] == 0 &]

N=10^9;
default(primelimit,N);
forprime(p=2,N, if (-2==Mod(2,p-1)^p, print1(p,", ")));
/* Joerg Arndt, Feb 06 2013 */

from sympy import primerange
A211349_list = [p for p in primerange(1,10**6) if p == 2 or pow(2,p,p-1) == p-3] # Chai Wah Wu, Mar 25 2021

a(19)-a(47) from Giovanni Resta, Feb 10 2013

a(48)-a(177) from Max Alekseyev, Jan 06 2018

A015945 Positive integers n such that n | (2^n + n/2 + 1).

Original entry on oeis.org

2, 10, 50, 250, 410, 1250, 2050, 5050, 6250, 10250, 16810, 25250, 31250, 51250, 84050, 126250, 156250, 207050, 256250, 336610, 405050, 420250, 510050, 631250, 689210, 781250, 1035250, 1281250, 1683050, 1750250, 2025250, 2101250, 2550250, 3156250, 3446050

Offset: 1

Robert G. Wilson v

nonn

A015945_list = [n for n in range(2,10**6,2) if pow(2,n,n) == n//2-1] # Chai Wah Wu, Mar 25 2021

a(n) = 2 * A015950(n). - Max Alekseyev, Aug 04 2011

A292330 Numbers k such that k^2 divides 4^k + 1.

Original entry on oeis.org

1, 5, 205, 168305, 2084645, 37217545, 1711493545, 2483072545, 2763736405, 8866165745, 30555604445, 55328770405, 169592124685, 378465215105, 423977184745, 2038602559445, 3815777985545, 7279122076645, 25250364710105, 28104435502445, 45424920502505, 55625535217765, 90160039460905

Offset: 1

Max Alekseyev, Sep 14 2017

nonn

Cf. A052539. Subsequence of A015950.

Cf. A127104, A128678.

A319222 Numbers k such that k divides 2^(2k+1) + 1.

Original entry on oeis.org

1, 3, 129, 2537, 51889, 101617, 226873, 270427, 653467, 945667, 1740979, 5819937, 6520987, 9828587, 15452867, 24950857, 51377539, 89519449, 108627601, 135776371, 160126609, 296338873, 310026163, 400431289, 641706243, 643359937, 678257563, 803419697, 902661523, 952431331, 1004273987, 1243893697, 1796055907

Offset: 1

Altug Alkan, Sep 13 2018

nonn

Also, numbers k such that 4^k == -1/2 (mod k) (cf. A296369). - Max Alekseyev, Sep 15 2018

If k is in the sequence, and m is another divisor of 2^(2*k+1)+1 and is coprime to k, then m*k is in the sequence. - Robert Israel, Sep 14 2018

Cf. A006517, A006521, A015950.

filter:= n -> 2 &^ (2*n+1)+1 mod n = 0:
select(filter, [$1..10^7]); # Robert Israel, Sep 14 2018

PARI
```
is_A319222(n) = Mod(2, n)^(2*n+1)==-1;
```

A015974 Numbers k that divide 4^k + 1, k not a power of 5.

Original entry on oeis.org

205, 1025, 2525, 5125, 8405, 12625, 25625, 42025, 63125, 103525, 128125, 168305, 202525, 210125, 255025, 315625, 344605, 517625, 640625, 841525, 875125, 1012625, 1050625, 1275125, 1578125, 1723025, 2042725, 2084645

Offset: 1

Robert G. Wilson v

nonn

Cf. A015950.

cp's OEIS Frontend

A015969 Numbers k that divide 16^k + 1.

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A327840 Numbers m that divide 4^m + 3.

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Magma

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A211349 Primes p such that p-1 divides 2^p + 2.

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A015945 Positive integers n such that n | (2^n + n/2 + 1).

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A292330 Numbers k such that k^2 divides 4^k + 1.

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A319222 Numbers k such that k divides 2^(2k+1) + 1.

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Maple

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A015974 Numbers k that divide 4^k + 1, k not a power of 5.

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