login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A033181 Absolute Euler pseudoprimes: composite n such that abs(a^((n-1)/2) mod n) = 1 for all a with (a,n) = 1. 3
1729, 2465, 15841, 41041, 46657, 75361, 162401, 172081, 399001, 449065, 488881, 530881, 656601, 670033, 838201, 997633, 1050985, 1615681, 1773289, 1857241, 2113921, 2433601, 2455921, 2704801, 3057601, 3224065, 3581761, 3664585, 3828001, 4463641, 4903921 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These numbers c have the property that for each prime divisor p, p-1 divides (c-1)/2. E.g., 2465 = 5*17*29; 1232/4 = 308; 1232/16 = 77; 1232/28 = 44. - Karsten Meyer, Nov 08 2005

All these numbers are Carmichael numbers (A002997). - Daniel Lignon, Sep 12 2015

LINKS

Daniel Lignon and Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 124 terms from Daniel Lignon)

Index entries for sequences related to pseudoprimes

MAPLE

filter:=  proc(n)

  local q;

  if isprime(n) then return false fi;

  if 2 &^ (n-1) mod n <> 1 then return false fi;

  if not numtheory:-issqrfree(n) then return false fi;

  for q in numtheory:-factorset(n) do

    if (n-1)/2 mod (q-1) <> 0 then return false fi

  od:

  true;

end proc:

select(filter, [seq(i, i=3..10^7, 2)]); # Robert Israel, Nov 24 2015

MATHEMATICA

absEulerpspQ[n_Integer?PrimeQ]:=False;

absEulerpspQ[n_Integer?EvenQ]:=False;

absEulerpspQ[n_Integer?OddQ]:=Module[{a=2},

While[a<n&&(GCD[a, n]!=1||!Unequal[PowerMod[a, (n-1)/2, n], 1, n-1]), a++];

(a==n)];

Select[Range[1, 1000000, 2], absEulerpspQ] (* Daniel Lignon, Sep 09 2015 *)

PROG

(Perl) use ntheory ":all"; my $n; foroddcomposites { say if is_carmichael($_) && vecall { (($n-1)>>1) % ($_-1) == 0 } factor($n=$_); } 1e6; # Dana Jacobsen, Dec 27 2015

CROSSREFS

Cf. A002997.

Sequence in context: A288153 A154717 A051388 * A198775 A154729 A083737

Adjacent sequences:  A033178 A033179 A033180 * A033182 A033183 A033184

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 11 1999

EXTENSIONS

"Absolute Euler pseudoprimes" added to name by Daniel Lignon, Sep 09 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)