A090088 -id:A090088 - cp's OEIS frontend

A090087 Smallest odd pseudoprimes to base n, not necessarily exceeding n. Compare with A007535 and A090086.

9, 341, 91, 15, 217, 35, 25, 9, 91, 9, 15, 65, 21, 15, 341, 15, 9, 25, 9, 21, 55, 21, 33, 25, 39, 9, 65, 9, 15, 49, 15, 25, 85, 15, 9, 35, 9, 39, 95, 39, 15, 205, 21, 9, 133, 9, 65, 49, 15, 21, 25, 51, 9, 55, 9, 15, 25, 57, 15, 341, 15, 9, 341, 9, 33, 65, 33, 25, 35, 69, 9, 85, 9, 15

Offset: 1

Labos Elemer, Nov 25 2003

nonn

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

Cf. A007535, A090986, A090088, A090089.

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

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

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

A090087 Smallest odd pseudoprimes to base n, not necessarily exceeding n. Compare with A007535 and A090086.

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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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