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A033553 3-Knödel numbers or D-numbers: numbers n > 3 such that n | k^(n-2)-k for all k with gcd(k, n) = 1. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Max Alekseyev, Oct 03 2016: (Start)

Also, composite numbers n such that A000010(p^k)=(p-1)*p^(k-1) divides n-3 for every prime power p^k dividing n (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^n == 8 (mod n) 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 n > 3 such that A002322(n) divides n-3. - Thomas Ordowski, Jul 15 2017

LINKS

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 489 terms from R. J. Mathar).

Eric Weisstein's World of Mathematics, D-Number.

Eric Weisstein's World of Mathematics, Knödel Numbers.

MAPLE

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(1000, 3) # Paolo P. Lava, Feb 24 2012

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

Cf. A002997, A050990, A050992, A050993, A208154-A208158, A277344.

Sequence in context: A175626 A096788 A050991 * A020192 A241809 A063174

Adjacent sequences:  A033550 A033551 A033552 * A033554 A033555 A033556

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

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

STATUS

approved

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Last modified August 22 03:28 EDT 2017. Contains 290942 sequences.