A015921 -id:A015921 - cp's OEIS frontend

A244673 Numbers k that divide 2^k + 4.

1, 2, 3, 4, 20, 260, 740, 2132, 2180, 5252, 43364, 49268, 49737, 80660, 130052, 293620, 542852, 661412, 717027, 865460, 1564180, 2185220, 2192132, 2816372, 3784916, 4377620, 4427540, 5722004, 6307652, 6919460, 8765252, 9084452, 9498260, 9723611, 11346260, 12208820, 12220132

Offset: 1

Derek Orr, Jul 14 2014

nonn

			2^2 + 4 = 8 is divisible by 2. Thus 2 is a term of this sequence.
2^3 + 4 = 12 is divisible by 3. Thus 3 is a term of this sequence.
2^4 + 4 = 20 is divisible by 4. Thus 4 is a term of this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..100
OEIS Wiki, 2^n mod n

The odd terms form A115976.

Cf. A015921, A140504.

A244673:=n->`if`(type((2^n+4)/n, integer), n, NULL): seq(A244673(n), n=1..10^5); # Wesley Ivan Hurt, Jul 15 2014
Alternative:
select(n -> 4 + 2&^n mod n = 0, [$1..10^5]); # Robert Israel, Jul 15 2014

Select[Range[1000], Mod[2^# + 4, #] == 0 &] (* Alonso del Arte, Jul 14 2014 *)
Join[{1,2,3},Select[Range[1223*10^4],PowerMod[2,#,#]==#-4&]] (* Harvey P. Dale, Jan 16 2023 *)

for(n=1, 10^8, if(Mod(2,n)^n+4==0, print1(n, ", "))) \\ Jens Kruse Andersen, Jul 15 2014

A276967 Odd integers n such that 2^n == 2^3 (mod n).

Original entry on oeis.org

1, 3, 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201, 213, 219, 237, 249, 267, 291, 303, 309, 315, 321, 327, 339, 381, 393, 399, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 693, 699, 717, 723, 731, 753, 771, 789, 807

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n - 3) == 1 (mod n).
Contains A033553 as a subsequence. Smallest term greater than 3 missing in A033553 is 731.
For all m, 2^A015921(m) - 1 belongs to this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015922.

Odd integers n such that 2^n == 2^k (mod n): A176997 (k = 1), A173572 (k = 2), this sequence (k = 3), A033984 (k = 4), A276968 (k = 5), A215610 (k = 6), A276969 (k = 7), A215611 (k = 8), A276970 (k = 9), A215612 (k = 10), A276971 (k = 11), A215613 (k = 12).

Join[{1}, Select[Range[1, 1023, 2], PowerMod[2, # - 3, #] == 1 &]] (* Alonso del Arte, Sep 22 2016 *)

isok(n) = (n % 2) && (Mod(2,n)^n==8); \\ Michel Marcus, Sep 23 2016

A296370 Numbers m such that 2^m == 3/2 (mod m).

Original entry on oeis.org

1, 111481, 465793, 79036177, 1781269903307, 250369632905747, 708229497085909, 15673900819204067

Offset: 1

Max Alekseyev, Dec 11 2017

Equivalently, 2^(m+1) == 3 (mod m).
Also, numbers m such that 2^(m+1) - 2 is a Fermat pseudoprime base 2, i.e., 2^(m+1) - 2 belongs to A015919 and A006935.
Some larger terms (may be not in order): 2338990834231272653581, 341569682872976768698011746141903924998969680637.

OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): this sequence (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Select[Range[10^6], Divisible[2^(# + 1) - 3, #] &] (* Robert Price, Oct 11 2018 *)

a(n) = A296104(n) - 1.

A303009 Numbers n such that both A002450(n)=(2^(2n)-1)/3 and A007583(n)=2*A002450(n)+1 are Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

23, 29, 41, 53, 89, 113, 131, 179, 191, 233, 239, 251, 281, 293, 341, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1271, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003

Offset: 1

Max Alekseyev, Apr 23 2018

nonn

It can be shown that if n is odd, it is a prime or a Fermat 4-pseudoprime (A020136) not divisible by 3. Similarly, 2n+1 is a prime or a Fermat 2-pseudoprime (A001567) not divisible by 3. In fact, the sequence is the union of the following six:
(i) primes n such that 2n+1 is prime (cf. A005384) and A007583(n) is composite, with smallest such term n=a(1)=23;
(ii) primes n==2 (mod 3) such that 2n+1 is a 2-psp (no such terms are known);
(iii) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is prime and A007583(n) is composite, with smallest such term n=a(15)=341;
(iv) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is 2-pseudoprime, with smallest such term n=268435455;
(v) n=2k, where 4k is in A015921 and k==1 (mod 3), such that 2n+1 is prime and A007583(n) is composite, with the smallest such term n=67166;
(vi) n=2k, where 4k is in A015921 and k==1 (mod 3), such that 2n+1 is a 2-psp, with the smallest such term n=9042986.

Cf. A002450, A007583, A175625, A175942, A300193, A303008, A303447, A303448.

Edited by Max Alekseyev, Aug 08 2019

A192109 Numbers k that divide 2^(k-1) - 2.

Original entry on oeis.org

1, 2, 6, 10, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 170, 178, 182, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526, 538, 542, 554, 562, 566, 586, 614, 622, 626

Offset: 1

Max Alekseyev, Apr 22 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Contains A216090 as subsequence.
Subsequence of A015921, consisting of the terms that are not multiples of 4.
The odd terms form A173572.
Cf. A014741, A000918.

import Data.List (elemIndices)
a192109 n = a192109_list !! (n-1)
a192109_list = map (+ 1) $ elemIndices 0 $ zipWith mod a000918_list [1..]
-- Reinhard Zumkeller, Apr 23 2013

Join[{1,2},Select[Range[700],PowerMod[2,#-1,#]==2&]] (* Harvey P. Dale, May 15 2015 *)

is(n)=Mod(2,n)^(n-1)==2 \\ Charles R Greathouse IV, Nov 04 2016

A319216 Numbers k such that k^2 + 1 divides 2^k + 2.

Original entry on oeis.org

0, 1, 3, 15, 79, 511, 4095, 6735, 65535, 2097151, 16777215, 75955411, 68719476735, 137438953471

Offset: 1

Altug Alkan, Sep 13 2018, following a suggestion from Max Alekseyev

Numbers t such that 2^t-1 is a term are 0, 1, 2, 4, 9, 12, 16, 21, ...
Primes p such that 2^((p^2-1)/2)-1 is a term are 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89, 103, 107, 127, 131, 139, 173, 191, 211, ...(cf. A062326).
a(14) > 10^11. - Hiroaki Yamanouchi, Sep 14 2018

Cf. A015921, A062326, A097958, A244673, A247220.

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

a(13) from Hiroaki Yamanouchi, Sep 14 2018

a(14) from Giovanni Resta, Sep 17 2018

A334634 Numbers m that divide 2^m + 11.

Original entry on oeis.org

1, 13, 16043199041, 91118493923, 28047837698634913

Offset: 1

Max Alekseyev, Sep 10 2020

Equivalently, numbers m such that 2^m == -11 (mod m).

No other terms below 10^17.

OEIS Wiki, 2^n mod n

Solutions to 2^n == k (mod n): A296370 (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), this sequence (k=-11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12).

a(5) from Sergey Paramonov, Oct 10 2021

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A244673 Numbers k that divide 2^k + 4.

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A276967 Odd integers n such that 2^n == 2^3 (mod n).

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A296370 Numbers m such that 2^m == 3/2 (mod m).

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A303009 Numbers n such that both A002450(n)=(2^(2n)-1)/3 and A007583(n)=2*A002450(n)+1 are Fermat pseudoprimes to base 2 (A001567).

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A192109 Numbers k that divide 2^(k-1) - 2.

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A319216 Numbers k such that k^2 + 1 divides 2^k + 2.

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A334634 Numbers m that divide 2^m + 11.

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