A090086 -id:A090086 - cp's OEIS frontend

A090098 Bases such that the smallest prime-power-pseudoprime is belonging to equals 25.

7, 18, 24, 32, 43, 51, 68, 74, 76, 99, 124, 126, 132, 151, 168, 174, 176, 182, 207, 218, 232, 243, 268, 274, 276, 282, 299, 318, 324, 326, 351, 374, 376, 382, 399, 407, 418, 426, 432, 443, 468, 474, 482, 499, 507, 518, 524, 526, 543, 551, 574, 576, 582, 599, 607

Offset: 1

Labos Elemer, Dec 01 2003

nonn

Values of x such that A090096(x) = 25.

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

Cf. A007535, A090086, A090096, A090097.

pspQ[n_,b_] := CompositeQ[n] &&  PowerMod[b, n - 1,n ] == 1 ; aQ[n_]:=pspQ[25, n] && AllTrue[{4,8,9,16}, !pspQ[#, n] &]; Select[Range[1000], aQ] (* Amiram Eldar, Sep 09 2019 *)

More terms from Amiram Eldar, Sep 09 2019

A250199 Smallest pseudoprime (>prime(n)) to base prime(n).

Original entry on oeis.org

341, 91, 124, 25, 15, 21, 45, 45, 33, 35, 49, 45, 105, 77, 65, 65, 87, 91, 85, 105, 111, 91, 105, 99, 105, 175, 133, 133, 117, 133, 153, 143, 148, 161, 175, 175, 186, 186, 231, 205, 185, 195, 217, 276, 231, 225, 217, 231, 285, 285, 259, 255, 363, 289, 301, 341, 286, 341, 322, 329

Offset: 1

Eric Chen, Feb 21 2015

nonn

Subsequence of A007535, see formula.

			a(7) = 45 because the 7th prime is 17, and the smallest pseudoprime (> 17) to base 17 is 45.

Eric Chen, Table of n, a(n) for n = 1..4096

Cf. A007535, A090086, A108162, A098650.

f[n_] := Block[{b = Prime[n], k = Prime[n] + 1}, While[PrimeQ[k] || PowerMod[b, k - 1, k] != 1, k++]; k]; Array[f, 60]

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

a(n) = A007535(A000040(n)).

A293512 Numbers k such that the smallest pseudoprime ( > k ) to base k, A007535(k), is a Carmichael number.

Original entry on oeis.org

348, 355, 358, 383, 388, 427, 448, 455, 478, 479, 485, 490, 491, 497, 499, 508, 509, 511, 515, 520, 521, 533, 535, 541, 545, 547, 551, 553, 556, 557, 559, 560, 679, 708, 759, 765, 777, 796, 807, 808, 822, 828, 838, 839, 847, 862, 891, 906, 928, 931, 933, 951

Offset: 1

Amiram Eldar, Oct 12 2017

nonn

			348 is the sequence since A007535(348) = 1105 is a Carmichael number.

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

Cf. A002997, A007535.

carmichaelQ[n_] := Divisible[n - 1, CarmichaelLambda[n]] && ! PrimeQ[n];
f[n_] := Block[{k = n}, While[GCD[n, k] > 1 || PrimeQ[k] || PowerMod[n, k - 1, k] != 1, j = k++]; k]; Select[Range[1000], carmichaelQ[f[#]] &] (* after Robert G. Wilson v at A090086 *)

A326614 Smallest Euler-Jacobi pseudoprime to base n.

Original entry on oeis.org

9, 561, 121, 341, 781, 217, 25, 9, 91, 9, 133, 91, 85, 15, 1687, 15, 9, 25, 9, 21, 221, 21, 169, 25, 217, 9, 121, 9, 15, 49, 15, 25, 545, 33, 9, 35, 9, 39, 133, 39, 21, 451, 21, 9, 481, 9, 65, 49, 25, 49, 25, 51, 9, 55, 9, 55, 25, 57, 15, 481, 15, 9, 529, 9, 33, 65, 33, 25, 35, 69, 9

Offset: 1

Richard N. Smith, Jul 14 2019

nonn

a(n) = 9 for n == 1 or 8 mod 9 (see A056020).

Richard N. Smith, Table of n, a(n) for n = 1..5000
Wikipedia, Euler-Jacobi pseudoprime

Cf. A047713, A048950, A090086 (least Fermat pseudoprime to base n), A298756 (least strong pseudoprime to base n).

ejpspQ[n_,b_] := CoprimeQ[n,b] && CompositeQ[n] && Mod[b^((n - 1)/2) - JacobiSymbol[b, n], n] == 0; leastEJpsp[b_] := Module[{k=9}, While[!ejpspQ[k, b], k+=2]; k]; Array[leastEJpsp, 100] (* Amiram Eldar, Jul 15 2019 *)

isok(k, n) = ((k%2==1) && (gcd(k, n)==1) && Mod(n, k)^((k-1)/2)==kronecker(n, k) && !isprime(k));
a(n) = my(k=2); while (! isok(k, n), k++); k; \\ Michel Marcus, Jul 15 2019

A371729 The number of pseudoprimes to base n that are smaller than n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 4, 0, 4, 0, 3, 1, 1, 0, 5, 3, 1, 2, 5, 0, 4, 1, 4, 3, 2, 1, 7, 0, 1, 1, 8, 0, 6, 2, 3, 3, 1, 0, 9, 2, 3, 1, 8, 0, 6, 3, 6, 1, 2, 0, 9, 3, 1, 7, 7, 1, 6, 2, 4, 1, 9, 0, 11, 2, 1, 7, 6, 1, 7, 3, 10, 5, 3, 0, 8, 4, 1, 1

Offset: 2

Amiram Eldar, Apr 05 2024

			a(2) = 0 since the smallest pseudoprime to base 2 (A001567) is 341 which is larger than 2.
a(5) = 1 since there is one pseudoprime to base 5 (A005936) that is smaller than 5: 4.
a(9) = 2 since there are 2 pseudoprimes to base 9 (A020138) that are smaller than 9: 4 and 8.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Wikipedia, Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A001567, A005936, A020138, A090086, A316504, A371730.

a[n_] := Count[Range[4, n-1], _?(CompositeQ[#] && PowerMod[n, # - 1, #] == 1 &)]; Array[a, 100, 2]

a(n) = {my(c = 0); forcomposite(k = 4, n-1, if(Mod(n, k)^(k-1) == 1, c++)); c;}

a(n) = 0 if and only if A090086(n) > n, or equivalently, n-1 is in A316504.

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).

A354689 Smallest Euler pseudoprime to base n.

Original entry on oeis.org

9, 341, 121, 341, 217, 185, 25, 9, 91, 9, 133, 65, 21, 15, 341, 15, 9, 25, 9, 21, 65, 21, 33, 25, 217, 9, 65, 9, 15, 49, 15, 25, 545, 21, 9, 35, 9, 39, 133, 39, 21, 451, 21, 9, 133, 9, 65, 49, 25, 21, 25, 51, 9, 55, 9, 33, 25, 57, 15, 341, 15, 9, 341, 9, 33, 65

Offset: 1

Jinyuan Wang, Jun 03 2022

nonn

An Euler pseudoprime to the base b is a composite number k which satisfies b^((k-1)/2) == +-1 (mod k).

Eric Weisstein's World of Mathematics, Euler Pseudoprime.

Cf. A006970, A090086, A298756, A326614.

a(n) = my(m); forcomposite(k=3, oo, if(k%2 && ((m=Mod(n, k)^(k\2))==1 || m==k-1), return(k)));

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(3, 2) if not isprime(k) and ((r:=pow(n, (k-1)//2, k)) == 1 or r == k-1))
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Jun 03 2022

a(n) = 9 for n == 1 or 8 (mod 9).

cp's OEIS Frontend

A090098 Bases such that the smallest prime-power-pseudoprime is belonging to equals 25.

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A250199 Smallest pseudoprime (>prime(n)) to base prime(n).

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A293512 Numbers k such that the smallest pseudoprime ( > k ) to base k, A007535(k), is a Carmichael number.

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A326614 Smallest Euler-Jacobi pseudoprime to base n.

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A371729 The number of pseudoprimes to base n that are smaller than n.

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

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A354689 Smallest Euler pseudoprime to base n.

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