A122780 -id:A122780 - cp's OEIS frontend

A153514 Terms of A122780 which are not Carmichael numbers A002997.

1, 6, 66, 91, 121, 286, 671, 703, 726, 949, 1541, 1891, 2665, 2701, 3281, 3367, 3751, 4961, 5551, 7107, 7381, 8205, 8401, 8646, 11011, 12403, 14383, 15203, 15457, 16471, 16531, 18721, 19345, 23521, 24046, 24661, 24727, 28009, 29161, 30857, 31621

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

For the intersection of this sequence and A153508, see A153513.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A002997, A122780, A153508, A153513.

filter:= proc(n) local p;
  if isprime(n) or (3 &^n - 3 mod n <> 0) then return false fi;
  if n::even then return true fi;
  if not numtheory:-issqrfree(n) then return true fi;
  for p in numtheory:-factorset(n) do
    if n-1 mod (p-1) <> 0 then return true fi
  od;
false
end proc:
filter(1):= true:
select(filter, [$1..10^5]); # Robert Israel, Jan 29 2017

okQ[n_] := !PrimeQ[n] && PowerMod[3, n, n] == Mod[3, n] && Mod[n, CarmichaelLambda[n]] != 1;
Select[Range[10^5], okQ] (* Jean-François Alcover, Mar 27 2019 *)

A005935 Pseudoprimes to base 3.

Original entry on oeis.org

91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 23521, 24046, 24661, 24727, 28009, 29161

Offset: 1

N. J. A. Sloane

nonn

Theorem: If q>3 and both numbers q and (2q-1) are primes then n=q*(2q-1) is a pseudoprime to base 3 (i.e. n is in the sequence). So for n>2, A005382(n)*(2*A005382(n)-1) is in the sequence (see Comments lines for the sequence A122780). 91,703,1891,2701,12403,18721,38503,49141... are such terms. This sequence is a subsequence of A122780. - Farideh Firoozbakht, Sep 13 2006
Composite numbers n such that 3^(n-1) == 1 (mod n).
Theorem (R. Steuerwald, 1948): if n is a pseudoprime to base b and gcd(n,b-1)=1, then (b^n-1)/(b-1) is a pseudoprime to base b. In particular, if n is an odd pseudoprime to base 3, then (3^n-1)/2 is a pseudoprime to base 3. - Thomas Ordowski, Apr 06 2016
Steuerwald's theorem can be strengthened by weakening his assumption as follows: if n is a weak pseudoprime to base b and gcd(n,b-1)=1, then ... - Thomas Ordowski, Feb 23 2021

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 91, p. 33, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, A12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..102839 (terms a(1)-a(798) from R. J. Mathar, a(799)-a(1000) from T. D. Noe)
J. Bernheiden, Pseudoprimes (Text in German)
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
F. Richman, Primality testing with Fermat's little theorem
Carlos Rivera, Conjecture 105: conjecture about twin primes and pseudo twin primes to base 3, The Prime Puzzles & Problems Connection.
Rudolf Steuerwald, Über die Kongruenz a^(n-1) == 1 (mod n), Sitzungsber. math.-naturw. Kl. Bayer. Akad. Wiss. München, 1948, pp. 69-70.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime
Index entries for sequences related to pseudoprimes

Pseudoprimes to other bases: A001567 (2), A005936 (5), A005937 (6), A005938 (7), A005939 (10).
Subsequence of A122780.
Cf. A005382.

base = 3; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Feb 21 2012 *)

is_A005935(n)={Mod(3,n)^(n-1)==1 & !ispseudoprime(n) & n>1}  \\ M. F. Hasler, Jul 19 2012

More terms from David W. Wilson, Aug 15 1996

A130433 Even pseudoprimes to base 3.

Original entry on oeis.org

286, 24046, 232726, 1304446, 1707266, 2232026, 3197806, 3922126, 4446982, 5603326, 5886166, 10123366, 10169926, 12304774, 13658086, 45133726, 47766286, 52249654, 62656126, 75421126, 76254046, 91459126, 91612246, 96956926, 108571606, 139868326, 151513846

Offset: 1

Alexander Adamchuk, May 26 2007, Jun 17 2007

nonn

The first 27 terms (halved) are given in Table 1 by Paszkiewicz and Rotkiewicz. - R. J. Mathar, Aug 22 2012

Amiram Eldar, Table of n, a(n) for n = 1..215 (terms below 10^12, calculated from the b-file at A005935; terms 1..68 from Jeppe Stig Nielsen)
Adam Paszkiewicz and Andrzej Rotkiewicz, On a Problem of H. J. A. Duprac, Tatra Mt. Math. Publ. 32 (2005) 15-32, MR2206908.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes.

Even terms of A005935.
Terms of A122780 that are congruent to 2 or 4 modulo 6.
Cf. A006935, A090082, A090083, A090084, A090085.
Cf. A130434, A130435, A130436, A130437, A130438, A130439, A130440, A130441, A130442, A130443.

Do[ f=PowerMod[ 3, 2n-1, 2n ]; If[ f==1, Print[ 2n ] ], {n,2,7000000} ]

forstep(n=4,10^10,2,Mod(3,n)^(n-1)==1 && print1(n,", ")) \\ Jeppe Stig Nielsen, Apr 25 2018

More terms from Alexander Adamchuk, Jun 17 2007

a(25)-a(27) from Jeppe Stig Nielsen, Apr 25 2018

A116629 Positive integers k such that 13^k == 3 (mod k).

Original entry on oeis.org

1, 2, 5, 166, 287603, 9241538, 2366680105, 8347156585, 21682897793, 6988245760865, 9045859950329, 10076294257985, 50299408064905, 254874726648713

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 24 2017

Some larger terms: 1440926367749746685, 76025040962646716305439353859479569558065. - Max Alekseyev, Jun 29 2011

Cf. A116609, A050259, A122780, A130422, A123061, A277554.

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), this sequence (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620 (k=10), A116638 (k=11), A116639 (k=15).

Join[{1, 2}, Select[Range[1, 5000], Mod[13^#, #] == 3 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1, 2}, Select[Range[10000000], PowerMod[13, #, #] == 3 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 3; \\ Michel Marcus, Nov 19 2017

Two more terms from Ryan Propper, Jan 09 2008

Terms 1,2 are prepended and a(9)-a(14) are added by Max Alekseyev, Jun 29 2011; Nov 24 2017

A153513 Composite numbers k such that 2^k-2 and 3^k-3 are both divisible by k and k is not a Carmichael number (A002997).

Original entry on oeis.org

2701, 18721, 31621, 49141, 83333, 83665, 88561, 90751, 93961, 104653, 107185, 176149, 204001, 226801, 228241, 276013, 282133, 534061, 563473, 574561, 622909, 653333, 665281

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Robert Israel)

Intersection of A153514 and A153508 (excluding the number 1).

Cf. A002997, A122780.

filter:= proc(n) local p;
  if isprime(n) or (2 &^n - 2 mod n <> 0) or (3 &^n - 3 mod n <> 0) then return false fi;
  if n::even then return true fi;
  if not numtheory:-issqrfree(n) then return true fi;
  for p in numtheory:-factorset(n) do
    if n-1 mod (p-1) <> 0 then return true fi
  od;
false
end proc:
select(filter, [$2..10^6]); # Robert Israel, Jan 29 2017

Reap[Do[If[CompositeQ[n] && Divisible[2^n-2, n] && Divisible[3^n-3, n] && Mod[n, CarmichaelLambda[n]] != 1, Print[n]; Sow[n]], {n, 2, 10^6}]][[2, 1]] (* Jean-François Alcover, Mar 25 2019 *)

A129493 Composite numbers k such that 3^k mod k is a power of 3.

Original entry on oeis.org

6, 10, 12, 14, 18, 22, 24, 26, 30, 33, 34, 36, 38, 39, 46, 51, 54, 56, 57, 58, 62, 63, 66, 69, 72, 74, 78, 82, 86, 87, 90, 91, 92, 93, 94, 99, 104, 106, 108, 111, 112, 116, 117, 118, 120, 121, 122, 123, 124, 129, 132, 134, 135, 141, 142, 144, 146, 148, 154, 158, 159

Offset: 1

Robert G. Wilson v, Apr 17 2007

Complement to composite numbers: 9, 15, 21, 25, 27, 28, 35, 42, 44, 45, 48, 49, 50, 52, 55, 60, 65, 68, 70, 75, ....

			14 is a member of the sequence since 3^14 mod 14 = 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A036236, A122780, A129492, A129494, A129495, A129496, A129497.

[k:k in [2..160]| not IsPrime(k)  and  not IsZero(a)  and (PrimeDivisors(a) eq [3]) where a is 3^k mod k ]; // Marius A. Burtea, Dec 04 2019

filter:= proc(n) local k;
  if isprime(n) then return false fi;
  k:= 3 &^ n mod n;
  k > 1 and k = 3^padic:-ordp(k,3)
end proc:
select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

Select[Range@ 161, IntegerQ@ Log[3, PowerMod[3, #, # ]] &]

A122784 Nonprimes k such that 7^k == 7 (mod k).

Original entry on oeis.org

1, 6, 14, 21, 25, 42, 105, 133, 231, 301, 325, 525, 561, 703, 817, 1105, 1729, 1825, 2101, 2353, 2465, 2821, 3277, 3325, 3486, 3913, 4011, 4525, 4825, 5565, 5719, 5901, 6601, 6697, 7525, 8321, 8911, 9331, 10225, 10325, 10585, 10621, 11041, 11521

Offset: 1

Farideh Firoozbakht, Sep 12 2006

nonn

Theorem: If both numbers q and 2q-1 are primes then q*(2q-1) is in the sequence iff q=2 or mod(q,14) is in the set {1, 5, 13}. 6, 703, 18721, 38503, 88831, 104653, 146611, 188191,... are such terms.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005938, A121014, A122780, A122781, A122782, A122783, A122785, A122786.

Select[Range[20000], ! PrimeQ[#] && PowerMod[7, #, #] == Mod[7, #] &]
With[{nn=12000},Select[Complement[Range[nn],Prime[Range[PrimePi[ nn]]]], PowerMod[7,#,#]==Mod[7,#]&]] (* Harvey P. Dale, Jul 12 2012 *)

A153515 Terms of A122782 which are not Carmichael numbers A002997.

Original entry on oeis.org

1, 4, 10, 15, 20, 65, 124, 190, 217, 310, 435, 781, 1541, 1891, 3565, 3820, 4123, 4495, 5461, 5611, 5662, 5731, 6735, 7449, 7813, 8029, 8290, 9881, 11041, 11476, 12801, 13021, 13333, 13981, 14981, 15751, 16297, 17767, 20345, 20710, 21361, 22791

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

Are there entries in this sequence which are also in A153513 ?

Yes. This subsequence starts 721801, 873181, 4504501, 8646121, 9006401, 9863461, 10403641, 10680265,... (similar to A153580). - R. J. Mathar, Mar 30 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002997, A122780, A153508, A153513.

Select[Range[10^4], !PrimeQ[#] && PowerMod[5, #, # ] == Mod[5, #] && Mod[#, CarmichaelLambda[#]] != 1 &] (* Amiram Eldar, Sep 19 2019 *)

A153580 Terms of A083737 which are not Carmichael numbers (A002997).

Original entry on oeis.org

721801, 873181, 4504501, 8646121, 9006401, 9863461, 10403641, 12322133, 14609401, 15913261, 18595801, 18736381, 20234341, 21397381, 22066201, 22369621, 22885129, 25326001, 25696133, 28312921, 36307981, 42702661

Offset: 1

Ray Chandler & Artur Jasinski, Dec 28 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2871 (calculated from the b-file at A083737)

Cf. A002997, A083737, A122780, A153508, A153513, A153514, A153515, A153519, A153581.

Select[Range[5*10^7], ! PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, # - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 && Mod[ #, CarmichaelLambda[ # ]] != 1 &] (* Ray Chandler, Dec 28 2008 *)

A153581 Pseudoprimes to bases 2,3,5 and 7 which are not Carmichael numbers (A002997).

Original entry on oeis.org

721801, 8646121, 10403641, 22885129, 36307981, 42702661, 46094401, 48064021, 52204237, 79398901, 80918281, 81954133, 114329881, 116151661, 143168581, 170782921, 188985961, 217145881, 220531501, 282707461, 299671921, 303373801, 326695141, 353815801, 361307521

Offset: 1

Ray Chandler & Artur Jasinski, Dec 28 2008

nonn

Terms congruent to 5 (mod 6): 468950021, 493108481, 659846021, 5936122901, 8144063621, ... - Robert G. Wilson v, Sep 03 2014

Terms not congruent to 1 (mod 12): 468950021, 493108481, 643767931, 659846021, 773131927, 5779230451, 5936122901, 7294056727, 8144063621, 9671001451, ... - Robert G. Wilson v, Sep 03 2014

Amiram Eldar, Table of n, a(n) for n = 1..1077 (terms 1..125 from Robert G. Wilson v)

Cf. A002997, A083737, A122780, A153508, A153513, A153514, A153515, A153519, A153580.

fQ[n_] := ! PrimeQ[n] && PowerMod[2, n - 1, n] == 1 && PowerMod[3, n - 1, n] == 1 && PowerMod[5, n - 1, n] == 1 && PowerMod[7, n - 1, n] == 1 && Mod[n, CarmichaelLambda[n]] != 1; Select[ Range[ 365000000], fQ] (* Ray Chandler, Dec 28 2008; corrected by Robert G. Wilson v, Sep 01 2014 *)

Terms a(8) and onward from Robert G. Wilson v, Sep 01 2014

cp's OEIS Frontend

A153514 Terms of A122780 which are not Carmichael numbers A002997.

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A005935 Pseudoprimes to base 3.

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A130433 Even pseudoprimes to base 3.

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A116629 Positive integers k such that 13^k == 3 (mod k).

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A153513 Composite numbers k such that 2^k-2 and 3^k-3 are both divisible by k and k is not a Carmichael number (A002997).

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Maple

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A129493 Composite numbers k such that 3^k mod k is a power of 3.

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Magma

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A122784 Nonprimes k such that 7^k == 7 (mod k).

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A153515 Terms of A122782 which are not Carmichael numbers A002997.

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