A033553 -id:A033553

A277344

3-Knödel numbers (A033553) that are not divisible by 3.

+20
4

50963, 5834755, 9835843, 155627923, 245056003, 332852435, 556268443, 724014203, 795650963, 831912763, 2440444163, 4080848203, 5067702643, 5140068643, 5555216803, 7461332483, 8438160643, 11766788323, 11951765003, 13058213003, 13483943203, 14528402983, 16644521435, 17847852803

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..775 (all terms below 10^16)

CROSSREFS

Intersection of A033553 and A242865.

Intersection of A033553 and A130133.

Subsequence of A015922.

KEYWORD

nonn

AUTHOR

Max Alekseyev, Oct 09 2016

STATUS

approved

A033554

In A015922, not in A033553.

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..42.

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

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

STATUS

approved

A050991

Duplicate of A033553.

+20
0

9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..20.

KEYWORD

dead

STATUS

approved

A208728

Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n.

+10
46

15, 35, 255, 455, 1295, 2703, 4355, 6479, 9215, 10439, 11951, 16211, 23435, 27839, 44099, 47519, 47879, 62567, 63167, 65535, 93023, 94535, 104195, 120959, 131327, 133055, 141155, 142883, 157079, 170819, 196811, 207935, 260831, 283679, 430199, 560735, 576719

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

GCD(b,n)=1 and b^(n+1) == 1 (mod n).

The sequence lists the squarefree composite numbers n such that every prime divisor p of n satisfies (p-1)|(n+1) (similar to Korselt's criterion).

The sequence can be considered as an extension of k-Knödel numbers to k negative, in this case equal to -1.

Numbers n > 3 such that b^(n+2) == b (mod n) for every integer b. Also, numbers n > 3 such that A002322(n) divides n+1. Are there infinitely many such numbers? It seems that such numbers n > 35 have at least three prime factors. - Thomas Ordowski, Jun 25 2017

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Carmichael Number

Eric Weisstein's World of Mathematics, Korselt's Criterion

Eric Weisstein's World of Mathematics, Knödel Numbers

EXAMPLE

6479 is part of the sequence because its prime factors are 11, 19 and 31: (6479+1)/(11-1)=648, (6479+1)/(19-1)=360 and (6479+1)/(31-1)=216.

MAPLE

with(numtheory); P:=proc(n) local d, ok, p;

if issqrfree(n) then p:=factorset(n); ok:=1;

for d from 1 to nops(p) do if frac((n+1)/(p[d]-1))>0 then ok:=0;

break; fi; od; if ok=1 then n; fi; fi; end: seq(P(i), i=5..576719);

MATHEMATICA

Select[Range[2, 576719], SquareFreeQ[#] && ! PrimeQ[#] && Union[Mod[# + 1, Transpose[FactorInteger[#]][[1]] - 1]] == {0} &] (* T. D. Noe, Mar 05 2012 *)

PROG

(PARI) is(n)=if(isprime(n)||!issquarefree(n)||n<3, return(0)); my(f=factor(n)[, 1]); for(i=1, #f, if((n+1)%(f[i]-1), return(0))); 1 \\ Charles R Greathouse IV, Mar 05 2012

CROSSREFS

Cf. A002322, A002997, A006972, A033553, A050990, A050992, A050993, A208154-A208158.

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Mar 01 2012

EXTENSIONS

Definition corrected by Thomas Ordowski, Jun 25 2017

STATUS

approved

A015922

Numbers k such that 2^k == 8 (mod k).

+10
20

1, 2, 3, 4, 8, 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201, 213, 219, 237, 248, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For all m, 2^A015921(m) - 1 belongs to this sequence.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..29055 (first 6822 terms from Zak Seidov)

OEIS Wiki, 2^n mod n.

MATHEMATICA

a015922Q[n_Integer] := If[Mod[2^n, n] == Mod[8, n], True, False];

a015922[n_Integer] :=

Flatten[Position[Thread[a015922Q[Range[n]]], True]];

a015922[1000000] (* Michael De Vlieger, Jul 16 2014 *)

m = 8; Join[Select[Range[m], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^3], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)

Join[{1, 2, 3, 4, 8}, Select[Range[650], PowerMod[2, #, #]==8&]] (* Harvey P. Dale, Aug 22 2020 *)

PROG

(PARI) isok(n) = Mod(2, n)^n == Mod(8, n); \\ Michel Marcus, Oct 13 2013, Jul 16 2014

CROSSREFS

Contains A033553 as a subsequence.

The odd terms form A276967.

Cf. A015921, A130133, A130134.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v

EXTENSIONS

First 5 terms inserted by David W. Wilson

STATUS

approved

A050990

2-Knödel numbers.

+10
17

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

MATHEMATICA

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

PROG

(PARI) a002322(n) = lcm(znstar(n)[2]);

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

A050992

4-Knödel numbers.

+10
13

6, 8, 12, 16, 20, 24, 28, 40, 44, 48, 52, 60, 68, 76, 80, 92, 112, 116, 120, 124, 148, 154, 164, 172, 188, 208, 212, 236, 240, 244, 264, 268, 280, 284, 292, 316, 332, 340, 356, 364, 388, 404, 412, 428, 436, 452, 508, 520, 524, 548, 556, 596, 604, 628, 652

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Knödel Numbers.

MATHEMATICA

Select[Range[6, 1000, 2], Divisible[# - 4, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

CROSSREFS

Cf. A002997, A050990, A033553, A050993, A208154-A208158.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

A050993

5-Knödel numbers.

+10
13

25, 65, 85, 145, 165, 185, 205, 265, 305, 365, 445, 485, 505, 545, 565, 685, 745, 785, 825, 865, 905, 965, 985, 1085, 1145, 1165, 1205, 1285, 1345, 1385, 1405, 1465, 1565, 1585, 1685, 1745, 1765, 1865, 1925, 1945, 1985, 2005, 2045, 2105, 2165

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

MATHEMATICA

Select[Range[10, 2500, 5], Divisible[# - 5, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

CROSSREFS

Cf. A002997, A050990, A033553, A050992, A208154-A208158.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

A208154

6-Knödel numbers.

+10
12

8, 10, 12, 18, 24, 30, 36, 42, 66, 72, 78, 84, 90, 102, 114, 126, 138, 168, 174, 186, 210, 222, 234, 246, 252, 258, 282, 318, 354, 366, 390, 396, 402, 426, 438, 456, 474, 498, 504, 534, 546, 582, 606, 618, 630, 642, 654, 678, 762, 786, 798, 822, 834, 894, 906

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Knödel Numbers.

MAPLE

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, 6);

MATHEMATICA

knodelQ[m_Integer?PrimeQ, n_Integer] := False; knodelQ[m_Integer, n_Integer] := Module[{i = n + 1}, While[i < m && (GCD[i, m] > 1 || Mod[i^(m - n), m] == 1), i++]; (i == m)]; Select[Range[1000], knodelQ[#, 6] &] (* Alonso del Arte, Feb 24 2012 *)

CROSSREFS

Cf. A002997, A050990, A033553, A050992, A050993, A208155-A208158.

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Feb 24 2012

STATUS

approved

A208158

10-Knödel numbers.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Knödel Numbers

MAPLE

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)

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Feb 24 2012

STATUS

approved