A050990 -id:A050990 - cp's OEIS frontend

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

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

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

A015921 Positive integers n such that 2^n == 4 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 132, 134, 142, 146, 158, 166, 170, 178, 182, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 372, 382, 386, 394, 398, 422, 446

Offset: 1

Robert G. Wilson v

nonn

Odd terms are given by A173572.

For all m, 2^A050259(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 A050990 as a subsequence.

Cf. A015910, A173572.

Select[Range[500], PowerMod[2, #, #] == 4 &] (* Alonso del Arte, Jul 07 2011 *)

Edited and terms 1,2,4 prepended by Max Alekseyev, Jul 29 2011

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 *)

A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Original entry on oeis.org

1122, 3458, 5642, 6734, 11102, 13202, 17390, 17822, 21170, 22610, 27962, 31682, 46002, 58682, 61778, 79730, 82082, 93314, 105266, 106262, 125490, 127946, 136202, 150722, 153254, 177122, 182002, 202202, 203870, 214370, 231842, 252434, 274298, 278462, 305102, 315282

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 2-Knödel numbers (A050990). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			1122 = 2*3*11*17 is squarefree and equals 10001100010_2, 1112120_3, 930_11, and 3f0_17 in base p = 2, 3, 11, and 17. Then s_2(1122) = 1+1+1+1 = 4 >= 2, s_3(1122) = 1+1+1+2+1+2 = 8 >= 3, s_11(1122) = 9+3 = 12 >= 11, and s_17(1122) = 3+f = 3+15 = 18 >= 17. Also, s_2(1122) = 4 == 2 (mod 1), s_3(1122) = 8 == 2 (mod 2), s_11(1122) = 12 == 2 (mod 10), and s_17(1122) = 18 == 2 (mod 16), so 1122 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2200
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A050990, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324405.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 2] &]

More terms from Amiram Eldar, Dec 05 2020

A208155 7-Knödel numbers.

Original entry on oeis.org

15, 49, 91, 133, 217, 259, 301, 427, 469, 511, 553, 679, 721, 763, 889, 973, 1015, 1057, 1099, 1141, 1267, 1351, 1393, 1477, 1561, 1603, 1687, 1897, 1939, 1981, 2107, 2149, 2191, 2317, 2359, 2443, 2569, 2611, 2653, 2779, 2863, 2947, 3031, 3073, 3199, 3241, 3409

Offset: 1

Paolo P. Lava, Feb 24 2012

nonn

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

Cf. A002997, A050990, A033553, A050992, A050993, A208154, A208156-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(10000,7);

(* First run program for A208154 to define knodelQ *) Select[Range[3500], knodelQ[#, 7] &] (* Alonso del Arte, Feb 24 2012 *)

A208157 9-Knödel numbers.

Original entry on oeis.org

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

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-A208156, 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(10000,9)

Select[Range[12, 1560, 3], Divisible[# - 9, CarmichaelLambda[#]]&] (* Jean-François Alcover, Mar 01 2018 *)

cp's OEIS Frontend

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

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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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A015921 Positive integers n such that 2^n == 4 (mod n).

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A050992 4-Knödel numbers.

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Mathematica

A050993 5-Knödel numbers.

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Mathematica

A208154 6-Knödel numbers.

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Maple

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A208158 10-Knödel numbers.

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A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

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