A033553 -id:A033553 - cp's OEIS frontend

A277344 3-Knödel numbers (A033553) that are not divisible by 3.

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

Offset: 1

Max Alekseyev, Oct 09 2016

nonn

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

Intersection of A033553 and A242865.
Intersection of A033553 and A130133.
Subsequence of A015922.

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

A050991 Duplicate of A033553.

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

Offset: 1

dead

A208728 Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Mar 01 2012

nonn

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
Proof that 15 and 35 are the only numbers with only two prime factors (and so all others have >= 3): Since n is squarefree and composite, it has at least two prime factors, p and q. If these are the only two, n = p*q. Then the criterion is that (p-1)|(n+1) -> (p-1)|pq+1, and q-1|pq+1. Write pq+1 = j*(p-1) = k*(q-1). Rearranging, p*(j-q)=j+1 and q*(k-p)=k+1. Since j = (pq+1)/(p-1), for large n, j~=q, and k~=p. But we see that p divides j+1~=q, and q divides k+1~=p. For large n this is only possible if p and q are roughly equal, so j-q=k-p=1. Otherwise, we have j+1 >= 2*p and k+1 >= 2*q, and which puts upper bounds on p and q. Enumerating these gives (p,q)=(3,5) and (p,q)=(5,7) as the only solutions. - Alex Meiburg, Oct 03 2024

			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.

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

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

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);

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

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

Definition corrected by Thomas Ordowski, Jun 25 2017

A015922 Numbers k such that 2^k == 8 (mod k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v

nonn

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

Michael De Vlieger, Table of n, a(n) for n = 1..29055 (first 6822 terms from Zak Seidov)
OEIS Wiki, 2^n mod n.

Contains A033553 as a subsequence.
The odd terms form A276967.
Cf. A015921, A130133, A130134.

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 *)

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

First 5 terms inserted by David W. Wilson

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

A050992 4-Knödel numbers.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

nonn

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

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

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

A050993 5-Knödel numbers.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

nonn

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

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

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

A208154 6-Knödel numbers.

Original entry on oeis.org

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

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, A208155-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(10000,6);

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 *)

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 *)

cp's OEIS Frontend

A277344 3-Knödel numbers (A033553) that are not divisible by 3.

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A033554 In A015922, not in A033553.

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A050991 Duplicate of A033553.

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A208728 Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n.

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A015922 Numbers k such that 2^k == 8 (mod k).

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A050990 2-Knödel numbers.

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Mathematica

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A050992 4-Knödel numbers.

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A050993 5-Knödel numbers.

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Mathematica

A208154 6-Knödel numbers.

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Maple

Mathematica

A208158 10-Knödel numbers.

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