A015919 -id:A015919 - cp's OEIS frontend

A015926 Positive integers n such that 2^n == 2^6 (mod n).

1, 2, 4, 6, 8, 10, 12, 16, 18, 24, 30, 31, 32, 36, 42, 48, 64, 66, 72, 78, 84, 90, 96, 102, 114, 126, 138, 144, 168, 174, 176, 186, 192, 210, 222, 234, 246, 252, 258, 282, 288, 318, 336, 354, 366, 390, 396, 402, 426, 438, 456, 474, 496, 498, 504, 510, 534, 546

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215610.

For all m, 2^A033981(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Contains A208154 as a subsequence.

Cf. A015919, A015921, A015922, A015924, A015925, A015927, A015929, A015931, A015932, A015935, A015937.

Select[Range[1000], Mod[2^# - 2^6, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)

Edited by Max Alekseyev, Jul 30 2011

A015929 Positive integers n such that 2^n == 2^8 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 32, 40, 48, 56, 60, 64, 80, 88, 96, 104, 120, 127, 128, 136, 140, 152, 160, 184, 192, 224, 232, 240, 248, 256, 260, 272, 296, 308, 320, 328, 344, 376, 384, 408, 416, 424, 472, 480

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215611.

For all m, 2^A051447(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Contains A208156 as a subsequence.

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015931, A015932, A015935, A015937.

Select[Range[1000], Mod[2^# - 2^8, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)

Edited by Max Alekseyev, Jul 30 2011

A015932 Positive integers n such that 2^n == 2^10 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 12, 16, 24, 28, 30, 32, 34, 48, 50, 64, 70, 73, 96, 110, 112, 128, 130, 150, 170, 190, 192, 230, 256, 290, 310, 330, 370, 384, 410, 430, 442, 448, 470, 512, 530, 532, 550, 590, 610, 670, 710

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215612.

For all m, 2^A033982(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Contains A208158 as a subsequence.

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015929, A015931, A015935, A015937.

Select[Range[1000], Mod[2^# - 2^10, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)

Edited by Max Alekseyev, Jul 30 2011

A015937 Positive integers n such that 2^n == 2^12 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 23, 24, 32, 36, 40, 42, 48, 60, 62, 64, 68, 72, 80, 84, 89, 96, 120, 126, 128, 132, 144, 156, 160, 168, 180, 192, 204, 228, 240, 252, 256, 276, 288, 312, 320, 336, 340, 348, 352, 360, 372, 384, 420, 444, 462

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215613.

For all m, 2^A051446(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015929, A015931, A015932, A015935.

With[{c=2^12},Select[Range[1,6000],Divisible[2^#-c,#]&]] (* Harvey P. Dale, Mar 20 2011 *)

Edited by Max Alekseyev, Jul 31 2011

A324050 Numbers satisfying Korselt's criterion: squarefree numbers n such that for every prime divisor p of n, p-1 divides n-1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

Offset: 1

Antti Karttunen, Feb 14 2019

Numbers n such that A002322(n) divides n-1.
Numbers that satisfy Korselt's criterion (see Mathworld and Wikipedia-links).
Union of {1}, primes and Carmichael numbers, A002997.
Differs from A008578 for the first time at term a(104) = 561, which (being a composite) is the first Carmichael number present.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Korselt's Criterion
Wikipedia, Carmichael numbers, Korselt's criterion

Union of A008578 and A002997. Intersection of A005117 and A087441. Subsequence of A015919.

Cf. A002322.

okQ[n_] := SquareFreeQ[n] && AllTrue[FactorInteger[n][[All, 1]], Divisible[n-1, #-1]&];
{1} ~Join~ Select[Range[2, 1000], okQ] (* Jean-François Alcover, Oct 22 2019 *)

isA324050(n) = if(1==n,1, my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); (1)); \\ After program Korselt in A002997
for(n=1,561,if(isA324050(n), print1(n,", ")));

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA324050(n) = (0==((n-1)%A002322(n)));

A277401 Positive integers n such that 7^n == 2 (mod n).

Original entry on oeis.org

1, 5, 143, 1133, 2171, 8567, 16805, 208091, 1887043, 517295383, 878436591673

Offset: 1

Seiichi Manyama, Oct 13 2016

All terms are odd.

No other terms below 10^15. Some larger terms: 181204957971619289, 21305718571846184078167, 157*(7^157-2)/1355 (132 digits). - Max Alekseyev, Oct 18 2016

			7 == 2 mod 1, so 1 is a term;
16807 == 2 mod 5, so 5 is a term.

Cf. A066438.
Cf. Solutions to 7^n == k (mod n): A277371 (k=-3), A277370 (k=-2), A015954 (k=-1), A067947 (k=1), this sequence (k=2), A277554 (k=3).
Cf. Solutions to b^n == 2 (mod n): A015919 (b=2), A276671 (b=3), A130421 (b=4), A124246 (b=5), this sequence (b=7), A116622 (b=13).

Join[{1},Select[Range[5173*10^5],PowerMod[7,#,#]==2&]] (* The program will generate the first 10 terms of the sequence; it would take a very long time to generate the 11th term. *) (* Harvey P. Dale, Apr 15 2020 *)

isok(n) = Mod(7, n)^n == 2; \\ Michel Marcus, Oct 13 2016

A066438(a(n)) = 2 for n > 1.

a(10) from Michel Marcus, Oct 13 2016

a(11) from Max Alekseyev, Oct 18 2016

A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

Original entry on oeis.org

12327121, 129816911251, 259621495381, 346157884801, 605767053061, 6317168754781, 6922923480721, 12634325182441, 18518799663001, 21634109682121, 24273469559431, 57114029344321, 65681131896901, 102718706568661, 135083316211741, 135818875521811, 153342494379361

Offset: 1

Felix Fröhlich, Sep 24 2014

nonn

Numbers k such that 2^k == 2 (mod k) and k is divisible by 3511^2.

Unless there are other Wieferich primes (A001220) besides 1093 and 3511, the intersection and the union of this sequence with A247830 are given by A219346 and A158358, respectively, and the even terms are given by A295740. - Max Alekseyev, Nov 26 2017 [The indices of the even terms in this sequence are 430, 525, 543, 701, 811, 826, 937, 1235, 1277, 1388, ... - Jianing Song, Feb 08 2019]

Jianing Song, Table of n, a(n) for n = 1..1470
R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473.
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.

Subsequence of each of (A001567 U A006935), A015919, A158358 composed of the terms divisible by 3511^2.

Cf. A001220, A013929, A219346, A247830, A295740.

vi=readvec("b158358.txt")
for(n=1, #vi, if(Mod(vi[n], 3511^2)==0, print1(vi[n], ", ")))

list(N)=select(k->Mod(2, k)^k==2, 3511^2*vector(N\3511^2\2, i, i)) \\ Jianing Song, Feb 07 2019

Name changed by Jianing Song, Feb 07 2019 to include the even pseudoprimes to base 2 (A006935) at the suggestion of Max Alekseyev.

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.

A235477 Primes whose base-2 representation also is the base-7 representation of a prime.

Original entry on oeis.org

2, 31, 47, 59, 103, 107, 173, 179, 181, 199, 211, 227, 229, 233, 367, 409, 443, 463, 487, 701, 743, 757, 823, 827, 877, 911, 919, 967, 1009, 1123, 1163, 1291, 1321, 1367, 1373, 1447, 1493, 1571, 1583, 1597, 1609, 1627, 1657, 1669, 1721, 1831, 1933, 1987

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
A subsequence of A027697, A015919, A197636 (conjectural).

			31 = 11111_2 and 11111_7 = 2801 are both prime, so 31 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235464 ⊂ A077721, A235475, A152079, A235266, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],7]]&] (* Harvey P. Dale, May 08 2021 *)

is(p,b=7)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A247830 Pseudoprimes to base 2 divisible by 1093^2.

Original entry on oeis.org

1194649, 3914864773, 5654273717, 6523978189, 22178658685, 26092328809, 31310555641, 41747009305, 53053167441, 58706246509, 74795779241, 85667085141, 237865367741, 333967711897, 467032496113, 575310702877, 601401837037, 962329192917, 1087566636885

Offset: 1

Felix Fröhlich, Sep 24 2014

nonn

Numbers k such that 2^k == 2 (mod k) and k is divisible by 1093^2. - Jianing Song, Apr 07 2019
In contrast to A247831, all terms of this sequence are odd. - Max Alekseyev, Nov 26 2017
Unless there are other Wieferich primes (A001220) besides 1093 and 3511, the intersection and the union of this sequence with A247831 are given by A219346 and A158358, respectively. - Max Alekseyev, Nov 26 2017

R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473.
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.

Subsequence of each of A001567, A015919, A158358 composed of the terms divisible by 1093^2.

Cf. A001220, A013929, A219346, A247831.

vi=readvec("b158358.txt")
for(n=1, #vi, if(Mod(vi[n], 1093^2)==0, print1(vi[n], ", ")))

cp's OEIS Frontend

A015926 Positive integers n such that 2^n == 2^6 (mod n).

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A015929 Positive integers n such that 2^n == 2^8 (mod n).

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A015932 Positive integers n such that 2^n == 2^10 (mod n).

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A015937 Positive integers n such that 2^n == 2^12 (mod n).

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A324050 Numbers satisfying Korselt's criterion: squarefree numbers n such that for every prime divisor p of n, p-1 divides n-1.

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A277401 Positive integers n such that 7^n == 2 (mod n).

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A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

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

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