A090085 -id:A090085 - cp's OEIS frontend

A130443 Even pseudoprimes to base 43.

6, 14, 42, 526974, 9157582, 21001206, 49419154, 156418318, 157058362, 223741702, 467016562, 531330346, 601692022, 681377698, 888739174, 931053466, 1037629198, 1390950926, 1392718618, 2175608494, 2377982166, 3045063946, 5136468646

Offset: 1

Alexander Adamchuk, May 26 2007, Jun 20 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..74 (terms below 10^11; terms 1..46 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A020171, A006935, A130433, A090082, A090083, A090084, A090085, A130434, A130435, A130436, A130437, A130438, A130439, A130440, A130441, A130442.

lst = {}; Do[ If[ PowerMod[43, 2n - 1, 2n] == 1, AppendTo[lst, 2n]], {n, 2, 1800000000}]; lst (* Robert G. Wilson v, Jun 01 2007 *)

is(k) = k > 2 && !(k % 2) &&  Mod(43, k)^(k-1) == 1; \\ Amiram Eldar, Sep 29 2024

More terms from Robert G. Wilson v, Jun 01 2007

A090083 Even pseudoprimes to base 9.

Original entry on oeis.org

4, 8, 28, 52, 286, 364, 532, 616, 946, 1036, 1288, 2806, 2926, 3052, 4376, 4636, 5356, 6364, 8744, 8866, 11476, 12124, 15964, 17446, 19096, 19684, 21196, 21736, 24046, 24388, 26596, 31876

Offset: 1

Labos Elemer, Nov 25 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Charles R Greathouse IV)
Index entries for sequences related to pseudoprimes.

Cf. A020138.

Cf. A006935, A130433, A090082, A090084, A090085, A130434, A130435, A130436, A130438, A130439, A130440, A130441, A130442, A130443.

Do[s=Mod[ -1+9^(n-1), n]; If[Equal[s, 0]&&!PrimeQ[n]&&EvenQ[n], Print[n]], {n, 1, 1000000}]

is(n)=Mod(9, n)^(n-1)==1&&!isprime(n)&&n%2==0 \\ Charles R Greathouse IV, Apr 12 2012

p=2; forprime(q=3, 1e8, forstep(n=p+1, q-1, 2, if(Mod(9, n)^(n-1)==1, print1(n", "))); p=q) \\ Charles R Greathouse IV, Apr 12 2012

A130441 Even pseudoprimes to base 37.

Original entry on oeis.org

4, 6, 12, 18, 28, 36, 66, 246, 268, 396, 1476, 1876, 2044, 2556, 2706, 3556, 5986, 9514, 11034, 16236, 17466, 25626, 31956, 34716, 120786, 149076, 153756, 246484, 259588, 281886, 283276, 483636, 552926, 559966, 623566, 670186, 721846, 846076, 1050666

Offset: 1

Alexander Adamchuk, May 26 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..548 (terms below 10^11; terms 1..243 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A020165, A006935, A130433, A090082, A090083, A090084, A090085, A130434, A130435, A130436, A130437, A130438, A130439, A130440, A130442, A130443.

Do[ f=PowerMod[ 37, 2n-1, 2n ]; If[ f==1, Print[ 2n ] ], {n,2,500000} ]
lst = {}; Do[ If[ PowerMod[37, 2n - 1, 2n] == 1, AppendTo[lst, 2n]], {n, 2, 2^31}]; lst (* Robert G. Wilson v, Jun 01 2007 *)

is(k) = k > 2 && !(k % 2) &&  Mod(37, k)^(k-1) == 1; \\ Amiram Eldar, Sep 29 2024

A130437 Even pseudoprimes to base 19.

Original entry on oeis.org

6, 18, 906, 5466, 257302, 825366, 1880082, 6637546, 6765826, 8936722, 9483706, 34087054, 51914026, 54806454, 57663334, 57819882, 67372378, 91835206, 98963734, 102985926, 117697186, 134457346, 143888806, 172530646, 206623266

Offset: 1

Alexander Adamchuk, May 26 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..145 (terms below 10^11; terms 1..51 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A020147 = Pseudoprimes to base 19. Cf. A006935 = Even pseudoprimes (or primes) to base 2: n divides 2^n - 2, n even. Cf. A130433 = Even pseudoprimes to base 3. Cf. A090082 = Even pseudoprimes to base 5. Cf. A090083, A090084, A090085. Cf. A130434, A130435, A130436, A130438, A130439, A130440, A130441, A130442, A130443.

Cf. A020147, A006935, A130433, A090082, A090083, A090084, A090085, A130434, A130435, A130436, A130438, A130439, A130440, A130441, A130442, A130443.

lst = {}; Do[ If[ PowerMod[19, 2n - 1, 2n] == 1, AppendTo[lst, 2n]], {n, 2, 2*10^9}]; lst (* Robert G. Wilson v, Jun 01 2007 *)

is(k) = k > 2 && !(k % 2) &&  Mod(19, k)^(k-1) == 1; \\ Amiram Eldar, Sep 29 2024

More terms from Robert G. Wilson v, Jun 01 2007

A090089 Smallest even pseudoprimes to odd base=4n-1, not necessarily exceeding n.

Original entry on oeis.org

286, 6, 10, 14, 6, 22, 26, 6, 34, 38, 6, 46, 10, 6, 58, 62, 6, 10, 74, 6, 82, 86, 6, 94, 14, 6, 106, 10, 6, 118, 122, 6, 10, 134, 6, 142, 146, 6, 14, 158, 6, 166, 10, 6, 178, 14, 6, 10, 194, 6, 202, 206, 6, 214, 218, 6, 226, 10, 6, 14, 22, 6, 10, 254, 6, 262, 14, 6, 274, 278, 6

Offset: 1

Labos Elemer, Nov 25 2003

nonn

There are no even pseudoprimes to an even base.

			n=1: base = 4n-1=3, smallest relevant power is -1+2^(286-1) which is divisible by 286.
Sieving further residue classes, smallest regularly arising pseudoprimes are 6,10 etc..

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

Cf. A007535, A090085, A090986, A090087, A090088.

a[n_] := Module[{k = 2}, While[GCD[n, k] > 1 || PrimeQ[k] || PowerMod[n, k - 1, k] != 1, k += 2]; k]; Table[a[4*n - 1], {n, 1, 100}] (* Amiram Eldar, Nov 11 2019 *)

a(n)=Min{x=4n-1 number; Mod[ -1+n^(x-1), x]=0}

A253233 Smallest even pseudoprime (>2n+1) in base 2n+1.

Original entry on oeis.org

4, 286, 124, 16806, 28, 70, 244, 742, 1228, 906, 1852, 154, 28, 286, 52, 66, 496, 442, 66, 1834, 344, 526974, 76, 506, 66, 70, 286, 1266, 2296, 946, 130, 5662, 112, 154, 14246, 370, 276, 8614, 2806, 2626, 112, 1558, 276, 2626, 19126, 1446, 322, 658, 176, 742, 190, 946, 5356, 742, 186, 190, 176, 8474, 2806, 2242, 148

Offset: 0

Eric Chen, May 17 2015

nonn

For an even base there are no even pseudoprimes.
Conjecture: There are infinitely many even pseudoprimes in every odd base.
Records: 4, 286, 16806, 526974, 815866, 838246, ..., and they occur at indices: 0, 1, 3, 21, 503, 691, ...

Eric Chen, Table of n, a(n) for n = 0..999 (a(0) corrected by _Georg Fischer_, Jan 20 2019)
Eric Weisstein's World of Mathematics, Fermat pseudoprime
Wikipedia, Fermat pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A005097, A090082, A090083, A090084, A090085, A090086, A090087, A090088, A090089, A108162, A130433, A130434, A130435, A130436, A130437, A130438, A130439, A130440, A139441, A130442, A130443, A007535.

f[n_] := Block[{k = 2 * n + 2}, While[PrimeQ[k] || OddQ[k] || PowerMod[2 * n + 1, k - 1, k] != 1, k++ ]; k]; Table[ f[n], {n, 0, 60}]

a(n) = for(k=n+1, 2^24, if(!isprime(2*k) && Mod(2*n+1, 2*k)^(2*k-1) == Mod(1, 2*k), return(2*k)))

a(A005097(n-1)) = A108162(n).

cp's OEIS Frontend

A130443 Even pseudoprimes to base 43.

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A090083 Even pseudoprimes to base 9.

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A130441 Even pseudoprimes to base 37.

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A130437 Even pseudoprimes to base 19.

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A090089 Smallest even pseudoprimes to odd base=4n-1, not necessarily exceeding n.

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A253233 Smallest even pseudoprime (>2n+1) in base 2n+1.

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Formula