A208157 -id:A208157 - cp's OEIS frontend

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.

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

A050990 2-Knödel numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 22, 24, 26, 30, 34, 38, 46, 56, 58, 62, 74, 82, 86, 94, 106, 118, 122, 132, 134, 142, 146, 158, 166, 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

Offset: 1

Eric W. Weisstein

nonn

Numbers k > 2 such that A002322(k) divides k-2. Contains all doubled primes and all doubled Carmichael numbers. - Thomas Ordowski, Apr 23 2017
Problem: are there infinitely many 2-Knodel numbers divisible by 4? - Thomas Ordowski, Jun 21 2017
Named after the Austrian mathematician and computer scientist Walter Knödel (1926-2018). - Amiram Eldar, Jun 08 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (first 690 terms from R. J. Mathar)
John H. Castillo and Jhony Fernando Caranguay Mainguez, The set of k-units modulo n, arXiv:1708.06812 [math.NT], 2017.
Eric Weisstein's World of Mathematics, Knödel Numbers.
Wikipedia, Knödel number.

Cf. A002997, A033553, A050992, A050993, A208154, A208155, A208156, A208157, A208158.

Select[Range[4, 460, 2], Divisible[# - 2, CarmichaelLambda@ #] &] (* Michael De Vlieger, Apr 24 2017 *)

a002322(n) = lcm(znstar(n)[2]);
forstep(n=4, 500, 2, if((n - 2)%a002322(n)==0, print1(n,", "))) \\ Indranil Ghosh, Jun 22 2017

A208158 10-Knödel numbers.

Original entry on oeis.org

12, 24, 28, 30, 50, 70, 110, 130, 150, 170, 190, 230, 290, 310, 330, 370, 410, 430, 442, 470, 530, 532, 550, 590, 610, 670, 710, 730, 790, 830, 890, 910, 970, 1010, 1030, 1070, 1090, 1130, 1270, 1310, 1370, 1390, 1490, 1510, 1570, 1630, 1650, 1670, 1730, 1790

Offset: 1

Paolo P. Lava, Feb 24 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Knödel Numbers

Cf. A002997, A050990, A033553, A050992, A050993, A208154-A208157.

with(numtheory);
knodel:=proc(i,k)
local a,n,ok;
for n from k+1 to i do
  ok:=1;
  for a from 1 to n do
     if gcd(a,n)=1 then  if (a^(n-k) mod n)<>1 then ok:=0; break; fi; fi;
  od;
  if ok=1 then print(n); fi;
od;
end:
knodel(10000,10)

Select[Range[12, 1790, 2], Divisible[# - 10, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

A015931 Positive integers n such that 2^n (mod n) == 2^9 (mod n).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 16, 17, 21, 27, 32, 45, 63, 64, 99, 105, 117, 124, 128, 153, 171, 189, 207, 254, 256, 261, 273, 279, 333, 369, 387, 423, 429, 477, 512, 513, 531, 549, 585, 603, 639, 657, 711, 747, 801, 873, 909, 927, 945, 963, 981, 1017, 1143, 1179, 1197, 1209, 1233, 1251, 1341, 1359, 1365, 1413, 1467, 1472, 1503, 1504, 1557, 1611, 1629, 1665, 1719, 1737, 1773, 1785, 1791, 1899, 1971

Offset: 1

Robert G. Wilson v

nonn

For all m, 2^A128123(m)-1 belongs to this sequence.

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

Contains A208157 as a subsequence.

The odd terms form A276970.

Select[Range[2000],PowerMod[2,9,#]==PowerMod[2,#,#]&] (* Harvey P. Dale, Apr 01 2017 *)

isok(n) = Mod(2, n)^n == 2^9; \\ Michel Marcus, Sep 23 2016

Edited by Max Alekseyev, Jul 30 2011

Definition clarified by Harvey P. Dale, Apr 01 2017

A276970 Odd integers n such that 2^n == 2^9 (mod n).

Original entry on oeis.org

1, 3, 5, 9, 17, 21, 27, 45, 63, 99, 105, 117, 153, 171, 189, 207, 261, 273, 279, 333, 369, 387, 423, 429, 477, 513, 531, 549, 585, 603, 639, 657, 711, 747, 801, 873, 909, 927, 945, 963, 981, 1017, 1143, 1179, 1197, 1209, 1233, 1251, 1341, 1359, 1365, 1413, 1467, 1503, 1557, 1611, 1629, 1665, 1719, 1737

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n-9) == 1 (mod n).
Contains A208157 as a subsequence.
For all m, 2^A128123(m)-1 belongs to this sequence.

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

The odd terms of A015931.

Odd integers n such that 2^n == 2^k (mod n): A176997 (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), this sequence (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).

m = 2^9; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^3, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 15 2018 *)

A208156 8-Knödel numbers.

Original entry on oeis.org

12, 14, 16, 20, 24, 32, 40, 48, 56, 60, 80, 88, 96, 104, 120, 136, 140, 152, 160, 184, 224, 232, 240, 248, 260, 296, 308, 328, 344, 376, 408, 416, 424, 472, 480, 488, 528, 536, 560, 568, 584, 632, 664, 680, 712, 728, 776, 808, 824, 856, 872, 904, 1016, 1040

Offset: 1

Paolo P. Lava, Feb 24 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Knödel Numbers

Cf. A002997, A050990, A033553, A050992, A050993, A208154, A208155, A208157, A208158.

with(numtheory);
knodel:=proc(i,k)
local a,n,ok;
for n from k+1 to i do
  ok:=1;
  for a from 1 to n do
     if gcd(a,n)=1 then  if (a^(n-k) mod n)<>1 then ok:=0; break; fi; fi;
  od;
  if ok=1 then print(n); fi;
od;
end:
knodel(10000,8)

Select[Range[10, 2000, 2], Divisible[# - 8, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

A265261 Smallest n-Knödel number, i.e., smallest composite c > n such that each b < c coprime to c satisfies b^(c-n) == 1 (mod c).

Original entry on oeis.org

561, 4, 9, 6, 25, 8, 15, 12, 21, 12, 15, 16, 33, 24, 21, 20, 65, 24, 51, 24, 45, 24, 33, 32, 69, 30, 39, 40, 65, 36, 87, 40, 45, 44, 51, 40, 85, 56, 57, 48, 65, 72, 91, 48, 63, 66, 69, 60, 141, 56, 63, 60, 65, 72, 75, 60, 63, 70, 87, 72, 133, 122, 93, 80, 165

Offset: 1

Felix Fröhlich, Apr 06 2016

nonn

Felix Fröhlich, Table of n, a(n) for n = 1..4000
Wikipedia, Knödel number.

Cf. A050990, A033553, A050992, A050993, A208154, A208155, A208156, A208157, A208158.

Table[SelectFirst[Range[n + 1, 10^3], Function[c, CompositeQ@ c && AllTrue[Range[1, c - 1] /. x_ /; ! CoprimeQ[x, c] -> Nothing, Mod[#^(c - n), c] == 1 &]]], {n, 65}] (* Michael De Vlieger, Apr 06 2016, Version 10 *)

a(n) = forcomposite(c=n+1, , my(i=0, j=0); for(b=1, c-1, if(gcd(b, c)==1, i++; if(Mod(b, c)^(c-n)==1, j++))); if(i==j, return(c)))

cp's OEIS Frontend

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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A050990 2-Knödel numbers.

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A208158 10-Knödel numbers.

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A015931 Positive integers n such that 2^n (mod n) == 2^9 (mod n).

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A276970 Odd integers n such that 2^n == 2^9 (mod n).

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A208156 8-Knödel numbers.

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A265261 Smallest n-Knödel number, i.e., smallest composite c > n such that each b < c coprime to c satisfies b^(c-n) == 1 (mod c).

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