

A033553


3Knödel numbers or Dnumbers: numbers n > 3 such that n  k^(n2)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
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OFFSET

1,1


COMMENTS

From Max Alekseyev, Oct 03 2016: (Start)
Also, composite numbers n such that A000010(p^k)=(p1)*p^(k1) divides n3 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


LINKS

R. J. Mathar, Table of n, a(n) for n = 1..489
Eric Weisstein's World of Mathematics, DNumber.
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^(nk) 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


PROG

(PARI) { isA033553(n) = my(p=factor(n)); for(i=1, matsize(p)[1], if( (n3)%eulerphi(p[i, 1]^p[i, 2]), return(0)); ); 1; } \\ Max Alekseyev, Oct 04 2016


CROSSREFS

Cf. A002997, A050990, A050992, A050993, A208154A208158, 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



