A015922 -id:A015922 - cp's OEIS frontend

A130134 Even terms in A015922.

2, 4, 8, 248, 17608, 90148, 106978, 253828, 364808, 1642798, 1926088, 6176264, 21879938, 30484408, 34634428, 96593698, 134396408, 223622468, 283585928, 327327388, 457961188, 809965148, 1709420344, 1815124028, 2164392968, 2360006456, 2431619908, 2500777828, 2922255548, 3888155428, 5672481928

Offset: 1

Zak Seidov, May 12 2007

nonn

Also, the even terms of A033554.

Max Alekseyev, Table of n, a(n) for n = 1..314 (all terms below 10^15)

Cf. A015922, A130133.

isok(n) = !(n % 2) && (Mod(2^n, n) == Mod(8, n)); \\ Michel Marcus, Oct 13 2013

a(8)-a(9) from Michel Marcus, Oct 13 2013

Terms a(10) onward from Max Alekseyev, Oct 11 2016

A130133 Terms in A015922 not divisible by 3.

Original entry on oeis.org

1, 2, 4, 8, 248, 731, 1333, 3503, 17608, 35003, 50963, 62611, 82603, 90148, 94643, 106978, 201295, 231311, 253828, 335723, 364808, 374573, 425323, 490915, 592595, 628015, 725203, 984343, 1031803, 1112023, 1136195, 1376903, 1411343, 1430003, 1642798, 1926088

Offset: 1

Zak Seidov, May 12 2007

nonn

Intersection of A015922 and A001651. - Michel Marcus, Oct 13 2013

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A001651, A015922, A130134.

Intersection with A033553 gives A277344.

a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1)) while
      irem(k, 3)=0 or 2&^k mod k <> 8 mod k do od; k
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 04 2014

{1, 2, 4, 8} ~Join~ Select[Range[2 10^6], PowerMod[2, #, #] == 8 && !Divisible[#, 3]&] (* Jean-François Alcover, Nov 02 2020 *)

isok(n) = (n % 3) && (Mod(2^n, n) == Mod(8, n)); \\ Michel Marcus, Oct 13 2013

a(17)-a(28) from Michel Marcus, Oct 13 2013

a(29)-a(36) from Alois P. Heinz, Jun 04 2014

A033554 In A015922, not in A033553.

Original entry on oeis.org

1, 2, 3, 4, 8, 248, 731, 1333, 1533, 2583, 2847, 3503, 4161, 4251, 4947, 4983, 5355, 5715, 6141, 7503, 8103, 9435, 9513, 9831, 10923, 12291, 13107, 14043, 17608, 17889, 20853, 23871, 25443, 27795, 29127, 29319, 29643, 30783, 32235, 33915, 34323, 35003

Offset: 1

David W. Wilson

nonn

Terms 1, 2, 3, 4, 8 prepended by Max Alekseyev, Oct 03 2016

A033553 3-Knödel numbers or D-numbers: numbers m > 3 such that m | k^(m-2)-k for all k with gcd(k, m) = 1.

Original entry on oeis.org

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, 753, 771, 789, 807, 813, 819

Offset: 1

David W. Wilson

nonn

From Max Alekseyev, Oct 03 2016: (Start)
Also, composite numbers m such that A000010(p^k)=(p-1)*p^(k-1) divides m-3 for every prime power p^k dividing m (cf. A002997).
Properties: (i) All terms are odd. (ii) A prime power p^k with k>1 may divide a term only if p=3 and k=2. (iii) Many terms are divisible by 3. The first term not divisible by 3 is a(2000) = 50963 (cf. A277344). (End)
All terms satisfy the congruence 2^m == 8 (mod m) and thus belong to A015922. Sequence a(n)/3 is nearly identical to A106317, which does not contain the terms 399/3 = 133 and 195/3 = 65. - Gary Detlefs, May 28 2014; corrected by Max Alekseyev, Oct 03 2016
Numbers m > 3 such that A002322(m) divides m-3. - Thomas Ordowski, Jul 15 2017
Called "D numbers" by Morrow (1951), in analogy to Carmichael numbers (A002997) that were also known then as "F numbers". Called "C_3 numbers" (and in general "C_k numbers") by Knödel (1953). Makowski (1962/63) proved that there are infinitely many k-Knödel numbers for all k >= 2. The 1-Knödel numbers are the Carmichael numbers (A002997). - Amiram Eldar, Mar 25 2024, Apr 21 2024

A. Makowski, Generalization of Morrow's D-Numbers, Bull. Belg. Math. Soc. Simon Stevin, Vol. 36 (1962/63), p. 71.
Paulo Ribenboim, The Little Book of Bigger Primes, 2nd ed., Springer, 2004, pp. 102-103.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 489 terms from R. J. Mathar).
John H. Castillo and Jhony Fernando Caranguay Mainguez, The set of k-units modulo n, Involve, a Journal of Mathematics, Vol. 15, No. 3 (2022), pp. 367-378; arXiv preprint, arXiv:1708.06812 [math.NT], 2017.
Walter Knödel, Carmichaelsche Zahlen, Math. Nachr., Vol. 9 (1953), pp. 343-350.
D. C. Morrow, Some Properties of D Numbers, The American Mathematical Monthly, Vol. 58, No. 5 (1951), pp. 329-330.
Eric Weisstein's World of Mathematics, D-Number.
Eric Weisstein's World of Mathematics, Knödel Numbers.
Wikipedia, Knödel number.

Cf. A002997, A050990, A050992, A050993, A208154, A208155, A208156, A208157, A208158, A277344.

isKnodel := proc(n,k)
    local a;
    for a from 1 to n do
        if igcd(a,n) = 1 then
            if modp(a&^(n-k),n) <> 1 then
                return false;
            end if;
        end if;
    end do:
    return true;
end proc:
isA033553 := proc(n)
    isKnodel(n,3) ;
end proc:
A033553 := proc(n)
    option remember;
    if n = 1 then
        return 9;
    else
        for a from procname(n-1)+1 do
            if isprime(a) then
                next;
            end if;
            if isA033553(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A033553(n),n=1..100) ; # R. J. Mathar, Aug 14 2024

Select[Range[4, 10^3], Divisible[# - 3, CarmichaelLambda[#]] &] (* Michael De Vlieger, Jul 15 2017 *)

{ isA033553(n) = my(p=factor(n)); for(i=1,matsize(p)[1], if( (n-3)%eulerphi(p[i,1]^p[i,2]), return(0)); ); 1; } \\ Max Alekseyev, Oct 04 2016

Edited by N. J. A. Sloane, May 07 2007

A128122 Numbers m such that 2^m == 6 (mod m).

Original entry on oeis.org

1, 2, 10669, 6611474, 43070220513807782

Offset: 1

Alexander Adamchuk, Feb 15 2007

No other terms below 10^17. - Max Alekseyev, Nov 18 2022

A large term: 862*(2^861-3)/281437921287063162726198552345362315020202285185118249390789 (203 digits). - Max Alekseyev, Sep 24 2016

			2 == 6 (mod 1), so 1 is a term;
4 == 6 (mod 2), so 2 is a term.

Joe K. Crump, 2^n mod n
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A000079 (k=0),A187787 (k=1/2), A296369 (k=-1/2), A006521 (k=-1), A296370 (k=3/2), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), this sequence (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)

Cf. A015910, A036236.

m = 6; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

1 and 2 added by N. J. A. Sloane, Apr 23 2007

a(5) from Max Alekseyev, Nov 18 2022

A295532 Positive integers n such that 13^n == 8 (mod n).

Original entry on oeis.org

1, 5, 371285, 3957661, 70348567451, 42831939409247

Offset: 1

Max Alekseyev, Nov 23 2017

No other terms below 10^15.

Cf. A116609, A127821, A015922, A277630.

Sequences 13^n == k (mod n): A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), this sequence (k=8), A116636 (k=9), A116620 (k=10), A116638 (k=11), A116639 (k=15).

Join[{1, 5}, Select[Range[4000000], PowerMod[13, #, #] == 8 &]] (* Robert Price, Apr 10 2020 *)

A296369 Numbers m such that 2^m == -1/2 (mod m).

Original entry on oeis.org

1, 5, 65, 377, 1189, 1469, 25805, 58589, 134945, 137345, 170585, 272609, 285389, 420209, 538733, 592409, 618449, 680705, 778805, 1163065, 1520441, 1700945, 2099201, 2831009, 4020029, 4174169, 4516109, 5059889, 5215769

Offset: 1

Max Alekseyev, Dec 10 2017

nonn

Equivalently, 2^(m+1) == -1 (mod m), or m divides 2^(m+1) + 1.

The sequence is infinite, see A055685.

Chai Wah Wu, Table of n, a(n) for n = 1..1192
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A296370 (k=3/2), A187787 (k=1/2), this sequence (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[10^5], Divisible[2^(# + 1) + 1, #] &] (* Robert Price, Oct 11 2018 *)

A296369_list = [n for n in range(1,10**6) if pow(2,n+1,n) == n-1] # Chai Wah Wu, Nov 04 2019

a(n) = A055685(n) - 1.

Incorrect term 4285389 removed by Chai Wah Wu, Nov 04 2019

A015927 Positive integers n such that 2^n == 2^7 (mod n).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 15, 16, 28, 32, 49, 62, 64, 91, 112, 128, 133, 196, 217, 255, 259, 301, 427, 448, 469, 511, 527, 553, 679, 721, 763, 784, 889, 973, 992, 1015, 1057, 1099, 1141, 1168, 1267, 1351, 1393, 1477, 1561, 1603, 1687, 1897, 1939, 1981, 2107, 2149, 2191, 2317, 2359, 2443, 2569, 2611, 2653

Offset: 1

Robert G. Wilson v

nonn

For all m, 2^A015922(m)-1 belongs to this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
OEIS Wiki, 2^n mod n

Contains A208155 as a subsequence.

The odd terms form A276969.

Select[Range[3000],PowerMod[2,#,#]==PowerMod[2,7,#]&] (* Harvey P. Dale, Mar 15 2018 *)

Edited by Max Alekseyev, Jul 30 2011

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

Original entry on oeis.org

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

cp's OEIS Frontend

A130134 Even terms in A015922.

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A130133 Terms in A015922 not divisible by 3.

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A033554 In A015922, not in A033553.

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A033553 3-Knödel numbers or D-numbers: numbers m > 3 such that m | k^(m-2)-k for all k with gcd(k, m) = 1.

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A128122 Numbers m such that 2^m == 6 (mod m).

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A295532 Positive integers n such that 13^n == 8 (mod n).

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A296369 Numbers m such that 2^m == -1/2 (mod m).

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A015927 Positive integers n such that 2^n == 2^7 (mod n).

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