A006970 -id:A006970 - cp's OEIS frontend

A262051 Euler pseudoprimes to base 3: composite integers such that abs(3^((n - 1)/2)) == 1 mod n.

121, 703, 1541, 1729, 1891, 2465, 2821, 3281, 4961, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009, 29341, 30857, 31621, 31697, 41041, 44287, 46657, 47197, 49141, 50881, 52633, 55969, 63139, 63973, 72041, 74593, 75361

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..64 from Daniel Lignon)

Cf. A006970 (base 2), this sequence (base 3), A001567 (base 4), A262052 (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[26000] + 1, eulerPseudoQ[#, 3] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(3, (2*n+1))^n == 1 ||  Mod(3, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262052 Euler pseudoprimes to base 5: composite integers such that abs(5^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

217, 781, 1541, 1729, 5461, 5611, 6601, 7449, 7813, 11041, 12801, 13021, 13333, 14981, 15751, 15841, 16297, 21361, 23653, 24211, 25351, 29539, 30673, 38081, 40501, 41041, 44173, 44801, 46657, 47641, 48133, 53971, 56033, 67921, 75361, 79381, 90241, 98173, 100651, 102311

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (term 1..65 from Daniel Lignon)

Cf. A006970 (base 2), A262051 (base 3), this sequence (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[27000] + 1, eulerPseudoQ[#, 5] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(5, (2*n+1))^n == 1 ||  Mod(5, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A048950 Base-3 Euler-Jacobi pseudoprimes.

Original entry on oeis.org

121, 703, 1729, 1891, 2821, 3281, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009, 29341, 31621, 41041, 44287, 46657, 47197, 49141, 50881, 52633, 55969, 63139, 63973, 74593, 75361, 79003, 82513

Offset: 1

Eric W. Weisstein

nonn

Odd composite k with gcd(k,3) = 1 and 3^((k-1)/2) == (3,k) (mod k) where (.,.) is the Jacobi symbol. - R. J. Mathar, Jul 15 2012

The base 5 Euler-Jacobi pseudoprimes are 781, 1541, 1729, 5461, 5611, 6601, 7449, ... - R. J. Mathar, Jul 15 2012 [Typo fixed; this is A375914. - Jianing Song, Sep 02 2024]

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Rotkiewicz, On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with Parameters L, Q in arithmetic progressions, Math. Comp 39 (159) (1982) 239-247.
Eric Weisstein's World of Mathematics, Euler-Jacobi Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A005935.
                                   |        b=2        |   b=3    |   b=5   |
-----------------------------------+-------------------+----------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
-----------------------------------+-------------------+----------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
-----------------------------------+-------------------+----------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490  | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
-----------------------------------+-------------------+----------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | this seq | A375914 |
       (union of first two)        |                   |          |         |
-----------------------------------+-------------------+----------+---------+
        Euler pseudoprimes         |      A006970      | A262051  | A262052 |
       (union of all three)        |                   |          |         |

Select[Range[1, 10^5, 2], GCD[#, 3] == 1 && CompositeQ[#] && PowerMod[3, (# - 1)/2, #] == Mod[JacobiSymbol[3, #], #] &] (* Amiram Eldar, Jun 28 2019 *)

is(n) = n%2==1 && gcd(n,3)==1 && Mod(3, n)^((n-1)/2)==kronecker(3,n)
forcomposite(c=1, 83000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Jul 15 2019

A033181 Absolute Euler pseudoprimes: odd composite numbers n such that a^((n-1)/2) == +-1 (mod n) for every a coprime to n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

These numbers n have the property that, for each prime divisor p, p-1 divides (n-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
These are odd composite numbers n such that b^((n-1)/2) == 1 (mod n) for every base b that is a quadratic residue modulo n and coprime to n. There are no odd composite numbers n such that b^((n-1)/2) == -1 (mod n) for every base b that is a quadratic non-residue modulo n and coprime to n. Note: the absolute Euler-Jacobi pseudoprimes do not exist. Theorem: if an absolute Euler pseudoprime n is a Proth number, then b^((n-1)/2) == 1 for every b coprime to n; by Proth's theorem. Such numbers are 1729, 8355841, 40280065, 53282340865, ...; for example, 1729 = 27*2^6 + 1 with 27 < 2^6. However, it seems that all absolute Euler pseudoprimes n satisfy the stronger congruence b^((n-1)/2) == 1 (mod n) for every b coprime to n, as evidenced by the modified Korselt's criterion (see the first comment). It should be noted that these are odd numbers n such that Carmichael's lambda(n) divides (n-1)/2. Also, these are odd numbers n > 1 coprime to Sum_{k=1..n-1} k^{(n-1)/2}. - Amiram Eldar and Thomas Ordowski, Apr 29 2019
Carmichael numbers k such that (p-1)|(k-1)/2 for each prime p|k. These are odd composite numbers k with half (the maximal possible fraction) of the numbers 1 <= b < k coprime to k that are bases in which k is an Euler-Jacobi pseudoprime, i.e. A329726((k-1)/2)/phi(k) = 1/2. - Amiram Eldar, Nov 20 2019
By Karsten Meyer's and Amiram Eldar's comment, this sequence is numbers k > 1 such that 2*psi(k) | (k-1), where psi = A002322. This means that if k is a term in this sequence, then we actually have a^((k-1)/2) == 1 (mod k) for every a coprime to k. - Jianing Song, Sep 03 2024

Daniel Lignon and Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 124 terms from Daniel Lignon)
Lorenzo Di Biagio, Euler Pseudoprimes for Half of the Bases, Integers, Vol. 12, No. 6 (2012), pp. 1231-1237, arXiv preprint, arXiv:1109.3596 [math.NT] (2011).
Math Help Forum, how many absolute euler pseudoprimes less than a million, Sep 2009.
Louis Monier, Evaluation and comparison of two efficient primality testing algorithms, Theoretical Computer Science, Vol. 11 (1980), pp. 97-108.
Index entries for sequences related to pseudoprimes

Cf. A002997, A006970, A047713, A080075, A329726.

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

absEulerpspQ[n_Integer?PrimeQ]:=False;
absEulerpspQ[n_Integer?EvenQ]:=False;
absEulerpspQ[n_Integer?OddQ]:=Module[{a=2},
While[aDaniel Lignon, Sep 09 2015 *)
aQ[n_] := Module[{f = FactorInteger[n], p},p=f[[;;,1]]; Length[p] > 1 && Max[f[[;;,2]]]==1 && AllTrue[p, Divisible[(n-1)/2, #-1] &]];Select[Range[3, 2*10^5], aQ] (* Amiram Eldar, Nov 20 2019 *)

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

a(n) == 1 (mod 4). - Thomas Ordowski, May 02 2019

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

Definition edited by Thomas Ordowski, Apr 29 2019

A306310 Odd numbers k > 1 such that 2^((k-1)/2) == -(2/k) = -A091337(k) (mod k), where (2/k) is the Jacobi (or Kronecker) symbol.

Original entry on oeis.org

341, 5461, 10261, 15709, 31621, 49981, 65077, 83333, 137149, 176149, 194221, 215749, 219781, 276013, 282133, 534061, 587861, 611701, 653333, 657901, 665333, 688213, 710533, 722261, 738541, 742813, 769757, 950797, 1064053, 1073021, 1109461, 1141141, 1357621, 1398101

Offset: 1

Jianing Song, Feb 06 2019

nonn

All terms are composite because for odd primes p we always have 2^((p-1)/2) == (2/p) = A091337(p) (mod p).
Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 5 (mod 8) such that 2^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]
Also numbers k in A001567 and congruent to 5 modulo 8 such that k - 1 divided by the multiplicative order of 2 modulo k is an even number.
Euler pseudoprimes (A006970) that are not Euler-Jacobi pseudoprimes (A047713). - Amiram Eldar, Oct 28 2019

			341 is a term because (2/341) = -1, and 2^((341-1)/2) == 1 (mod 341).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

Cf. A091337, A001567, A007814.
                                   |        b=2        |   b=3   |   b=5   |
-----------------------------------+-------------------+---------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915 |
-----------------------------------+-------------------+---------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916 |
-----------------------------------+-------------------+---------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      this seq     | A375490 | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |         |
-----------------------------------+-------------------+---------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914 |
       (union of first two)        |                   |         |         |
-----------------------------------+-------------------+---------+---------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052 |
       (union of all three)        |                   |         |         |

isA306310(k)=(k%8==5) && Mod(2, k)^((k-1)/2)==1

isok(k) = (k>1) && (k%2) && (Mod(2, k)^((k-1)/2) == Mod(-kronecker(2, k), k)); \\ Michel Marcus, Feb 07 2019

A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).

Original entry on oeis.org

1541, 2465, 4961, 30857, 31697, 72041, 83333, 162401, 192713, 206981, 258017, 359369, 544541, 565001, 574397, 653333, 929633, 1018601, 1032533, 1133441, 1351601, 1373633, 1904033, 1953281, 2035661, 2797349, 2864501, 3264797, 3375041, 3554633, 3562361, 3636161

Offset: 1

Jianing Song, Sep 01 2024

nonn

Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 5 (mod 12) such that 3^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]

			1541 is a term because (3/1541) = -1, and 3^((1541-1)/2) == 1 (mod 1541).

Jianing Song, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

                                   |        b=2        |   b=3    |   b=5   |
-----------------------------------+-------------------+----------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
-----------------------------------+-------------------+----------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
-----------------------------------+-------------------+----------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | this seq | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
-----------------------------------+-------------------+----------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950  | A375914 |
       (union of first two)        |                   |          |         |
-----------------------------------+-------------------+----------+---------+
        Euler pseudoprimes         |      A006970      | A262051  | A262052 |
       (union of all three)        |                   |          |         |

isA375490(k) = (k>1) && gcd(k,6)==1 && Mod(3,k)^((k-1)/2)==-kronecker(3,k)

isA375490(k) = k%12==5 && Mod(3,k)^((k-1)/2)==1 \\ Jianing Song, Sep 07 2024

A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).

Original entry on oeis.org

217, 13333, 16297, 23653, 30673, 44173, 46657, 48133, 56033, 98173, 130417, 131977, 136137, 179893, 188113, 190513, 197633, 267977, 334153, 334657, 347777, 360533, 407353, 412933, 421637, 486157, 667153, 670033, 677917, 694153, 710533, 765073, 839833, 935137, 997633

Offset: 1

Jianing Song, Sep 01 2024

nonn

Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 13, 17 (mod 20) such that 5^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]

			217 is a term because (5/217) = -1, and 5^((217-1)/2) == 1 (mod 217).

Jianing Song, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | this seq |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375816(k) = (k>1) && gcd(k,10)==1 && Mod(5,k)^((k-1)/2)==-kronecker(5,k)

isA375816(k) = (k%20==13 || k%20==17) && Mod(5,k)^((k-1)/2)==1

A375914 Base-5 Euler-Jacobi pseudoprimes: odd composite k coprime to 5 such that 5^((k-1)/2) == (5/k) (mod n), where (5/k) is the Jacobi symbol (or Kronecker symbol).

Original entry on oeis.org

781, 1541, 1729, 5461, 5611, 6601, 7449, 7813, 11041, 12801, 13021, 14981, 15751, 15841, 21361, 24211, 25351, 29539, 38081, 40501, 41041, 44801, 47641, 53971, 67921, 75361, 79381, 90241, 100651, 102311, 104721, 106201, 106561, 112141, 113201, 115921, 121463, 133141

Offset: 1

Jianing Song, Sep 02 2024

nonn

			781 is a term because 781 = 11*71 is composite, (5/781) = 1, and 5^((781-1)/2) == 1 (mod 781).
7813 is a term because 7813 = 13*601 is composite, (5/7813) = -1, and 5^((7813-1)/2) == -1 (mod 7813).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | this seq |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375914(k) = k>1 && !isprime(k) && gcd(k,10)==1 && Mod(5,k)^((k-1)/2)==kronecker(5,k)

A375915 Composite numbers k == 1, 9 (mod 10) such that 5^((k-1)/2) == 1 (mod k).

Original entry on oeis.org

781, 1541, 1729, 5461, 5611, 6601, 7449, 11041, 12801, 13021, 14981, 15751, 15841, 21361, 24211, 25351, 29539, 38081, 40501, 41041, 44801, 47641, 53971, 67921, 75361, 79381, 90241, 100651, 102311, 104721, 106201, 106561, 112141, 113201, 115921, 133141, 135201, 141361

Offset: 1

Jianing Song, Sep 02 2024

nonn

Odd composite numbers k such that 5^((k-1)/2) == (5/k) = 1 (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol).

			29539 is a term because 29539 = 109*271 is composite, 29539 == 9 (mod 10), and 5^((29539-1)/2) == 1 (mod 29539).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | this seq |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375915(k) = (k>1) && !isprime(k) && (k%10==1 || k%10==9) && Mod(5,k)^((k-1)/2) == 1

A375916 Composite numbers k == 3, 7 (mod 10) such that 5^((k-1)/2) == -1 (mod k).

Original entry on oeis.org

7813, 121463, 195313, 216457, 315283, 319507, 353827, 555397, 559903, 753667, 939727, 1164083, 1653667, 1663213, 1703677, 1809697, 1958503, 2255843, 2339377, 2423323, 2942333, 2987167, 3313643, 4265257, 4635053, 5376463, 5979247, 6611977, 7784297, 7859707

Offset: 1

Jianing Song, Sep 02 2024

nonn

Odd composite numbers k such that 5^((k-1)/2) == (5/k) = -1 (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol).

			216457 is a term because 216457 = 233*929 is a composite, 216457 == 7 (mod 10), and 5^((216457-1)/2) == -1 (mod 216457).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | this seq |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375916(k) = !isprime(k) && (k%10==3 || k%10==7) && Mod(5,k)^((k-1)/2) == -1

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A262051 Euler pseudoprimes to base 3: composite integers such that abs(3^((n - 1)/2)) == 1 mod n.

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A262052 Euler pseudoprimes to base 5: composite integers such that abs(5^((n - 1)/2)) == 1 mod n.

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Mathematica

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A048950 Base-3 Euler-Jacobi pseudoprimes.

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Mathematica

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A033181 Absolute Euler pseudoprimes: odd composite numbers n such that a^((n-1)/2) == +-1 (mod n) for every a coprime to n.

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A306310 Odd numbers k > 1 such that 2^((k-1)/2) == -(2/k) = -A091337(k) (mod k), where (2/k) is the Jacobi (or Kronecker) symbol.

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A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).

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A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).

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PARI

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A375914 Base-5 Euler-Jacobi pseudoprimes: odd composite k coprime to 5 such that 5^((k-1)/2) == (5/k) (mod n), where (5/k) is the Jacobi symbol (or Kronecker symbol).

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