A020138 -id:A020138 - cp's OEIS frontend

A090083 Even pseudoprimes to base 9.

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

A020235 Strong pseudoprimes to base 9.

Original entry on oeis.org

91, 121, 671, 703, 1541, 1729, 1891, 2821, 3281, 3367, 3751, 5551, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 16471, 16531, 18721, 19345, 23521, 24661, 24727, 28009, 29341, 30857, 31621, 32791, 38503, 44287, 46999, 47197, 49051, 49141, 53131

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..369 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A020138, A020229 (base 3), A020307 (base 81).

A262154 Pseudoprimes to base 9, written in base 9.

Original entry on oeis.org

4, 8, 31, 57, 111, 144, 247, 347, 444, 627, 651, 754, 825, 854, 861, 1261, 1264, 1371, 1457, 1681, 1811, 2102, 2331, 2531, 3338, 3378, 3581, 3631, 3757, 3774, 4011, 4161, 4445, 4551, 5127, 6002, 6321, 6722, 7311, 7547, 8651, 10044, 10101, 10637, 11111, 11762, 12464, 12831, 12885, 13141, 13201, 15461, 16084, 16451

Offset: 1

Abdul Gaffar Khan, Sep 13 2015

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

Cf. A007095 (numbers in base 9), A020138 (pseudoprimes to base 9).

Cf. A258189, A262101, A262102, A262103, A262104, A262105.

base = 9; t = {}; n = 1;
While[Length[t] < 80, n++;
If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1,
  AppendTo[t, FromDigits@IntegerDigits[n, 9]]]]; t

lista(nn, b=9) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007095(A020138(n)).

A122786 Nonprimes n such that 9^n == 9 (mod n).

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 15, 18, 24, 28, 36, 45, 52, 66, 72, 91, 121, 153, 205, 276, 286, 364, 366, 369, 396, 435, 511, 532, 561, 616, 671, 697, 703, 726, 804, 946, 949, 1035, 1036, 1105, 1128, 1288, 1387, 1541, 1729, 1737, 1845, 1854, 1891, 2196, 2465, 2501, 2556, 2665

Offset: 1

Farideh Firoozbakht, Sep 12 2006

nonn

Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) then 9^n==9 (mod n) (n is in the sequence). So A005382*(2*A005382-1)= 6,15,91,703,1891,2701,12403,18721,... is the related subsequence. A020138 is a subsequence of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A005382, A020138.

q:= n-> is(not isprime(n) and (9 &^ n mod n) = (9 mod n)):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 06 2019

Select[Range[4000], ! PrimeQ[ # ] && Mod[9^#, # ] == Mod[9, # ] &]
Join[{1,4,6,8,9},Select[Range[3000],CompositeQ[#]&&PowerMod[9,#,#]==9&]] (* Harvey P. Dale, Jul 17 2014 *)

isok(n) = !isprime(n) && (Mod(9,n)^n == Mod(9, n)); \\ Michel Marcus, Mar 06 2019

A306448 Pseudoprimes to base 9 that are not squarefree.

Original entry on oeis.org

4, 8, 28, 52, 121, 364, 532, 616, 1036, 1288, 3052, 3751, 4376, 4636, 4961, 5356, 6364, 7381, 8744, 11011, 11476, 12124, 15964, 19096, 19684, 21196, 21736, 24388, 26596, 29161, 31876, 32791, 37576, 40132, 45676, 47972, 53092, 61831, 67276, 72136, 80476, 80956, 86296

Offset: 1

Jianing Song, Feb 16 2019

nonn

Numbers k that are not squarefree and satisfy 9^(k-1) == 1 (mod k).
Any term is divisible by the square of a base-9 Wieferich prime ({2} U {base-3 Wieferich primes} = {2} U A014127 = {2, 11, 1006003, ...}).
Intersection of A020138 and A013929.

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

Pseudoprimes to base b that are not squarefree: A158358 (b=2), A244065 (b=3), A243010 (b=5), A243089 (b=7), A243090 (b=8), this sequence (b=9), A306449 (b=10).

Cf. also A014127, A020138, A013929.

for(n=1, 10^5, if(Mod(9, n)^(n-1)==1 && !issquarefree(n), print1(n, ", ")))

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.

A263239 Euler pseudoprimes to base 9: composite integers such that abs(9^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

4, 28, 91, 121, 286, 532, 671, 703, 949, 1036, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4636, 4961, 5551, 6364, 6601, 7381, 8401, 8911, 10585, 11011, 11476, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 19684, 23521, 24046, 24661, 24727

Offset: 1

Daniel Lignon, Oct 12 2015

nonn

Even numbers are permitted since 9 is an integer square. - Charles R Greathouse IV, Oct 12 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..116 from Daniel Lignon)

Cf. A020138 (pseudoprimes to base 9).

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudo9Q[n_]:=(Mod[9^((n-1)/2)+1,n]==0 ||Mod[9^((n-1)/2)-1,n]==0) && Not[PrimeQ[n]];
Select[Range[2,200000],eulerPseudo9Q]

is(n) = abs(centerlift(Mod(3, n)^(n-1)))==1 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 12 2015

A247906 a(n) = n-th pseudoprime to base n.

Original entry on oeis.org

561, 286, 341, 781, 1105, 1105, 133, 364, 703, 793, 1105, 1099, 1891, 6541, 1271, 3991, 1649, 1849, 3059, 7363, 2047, 1738, 4537, 1128, 3145, 2993, 5365, 4069, 4097, 7421, 2465, 11305, 2937, 16589, 4495, 2044, 6601, 26885, 13073, 6892, 22945, 3885, 8695, 10879

Offset: 2

Felix Fröhlich, Sep 26 2014

nonn

			a(2) = A001567(2) = 561.
a(3) = A005935(3) = 286.

Cf. A007535, A098650.

Cf. Pseudoprimes to base b: A001567 (b=2), A005935 (b=3), A020136 (b=4), A005936 (b=5), A005937 (b=6), A005938 (b=7), A020137 (b=8), A020138 (b=9).

for(n=2, 20, i=0; forcomposite(c=2, 1e9, if(Mod(n, c)^(c-1)==1, i++; if(i==n, print1(c, ", "); i=0; break({1}))); if(c==1e9, print1(">1e9, "))))

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A090083 Even pseudoprimes to base 9.

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A020235 Strong pseudoprimes to base 9.

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A262154 Pseudoprimes to base 9, written in base 9.

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A122786 Nonprimes n such that 9^n == 9 (mod n).

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A306448 Pseudoprimes to base 9 that are not squarefree.

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

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A263239 Euler pseudoprimes to base 9: composite integers such that abs(9^((n - 1)/2)) == 1 mod n.

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A247906 a(n) = n-th pseudoprime to base n.

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