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

Author

David W. Wilson

Keywords

• nonn

Comments

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

References

• 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.

Links

• 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.

Crossrefs

Cf. A002997, A050990, A050992, A050993, A208154, A208155, A208156, A208157, A208158, A277344.

Programs

Maple

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

Mathematica

Select[Range[4, 10^3], Divisible[# - 3, CarmichaelLambda[#]] &] (* Michael De Vlieger, Jul 15 2017 *)

PARI

{ 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

Extensions

Edited by N. J. A. Sloane, May 07 2007