A277344 -id:A277344 - 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

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

A242865 Numbers n such that 3^(n - 3) is congruent to 1 modulo n.

Original entry on oeis.org

3, 9299, 31903, 50963, 87043, 115918, 116891, 219827, 241043, 394243, 550243, 617503, 760243, 806623, 1029253, 1050787, 1458083, 1642798, 1899458, 2864755, 3205387, 3588115, 3839363, 4164578, 5041223, 5610583, 5834755, 5977555, 7837903, 8005558, 8067433, 8128823, 9007603, 9298903, 9449113, 9617443, 9835843

Offset: 1

Felix Fröhlich, May 24 2014

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..372 (terms 1..163 from Felix Fröhlich)

Cf. A067945, A173572.

Intersection with A033553 gives A277344.

Select[Range[10^4], Mod[3^(# - 3), #] == 1 &] (* Alonso del Arte, May 27 2014 *)

for(n=3, 10^6, if(Mod(3, n)^(n-3)==1, print1(n, ", ")))

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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A130133 Terms in A015922 not divisible by 3.

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Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A242865 Numbers n such that 3^(n - 3) is congruent to 1 modulo n.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI