OFFSET
1,1
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 07 2007
STATUS
approved