A262051 -id:A262051 - cp's OEIS frontend

A262053 Euler pseudoprimes to base 6: composite integers such that abs(6^((n - 1)/2)) == 1 mod n.

185, 217, 301, 481, 1111, 1261, 1333, 1729, 2465, 2701, 3421, 3565, 3589, 3913, 5713, 6533, 8365, 10585, 11041, 11137, 12209, 14701, 15841, 17329, 18361, 20017, 21049, 22049, 29341, 31021, 31621, 34441, 36301, 38081, 39305, 39493, 41041, 43621, 44801, 46657

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..86 from Daniel Lignon)

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), this sequence (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[25000] + 1, eulerPseudoQ[#, 6] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(6, (2*n+1))^n == 1 ||  Mod(6, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262054 Euler pseudoprimes to base 7: composite integers such that abs(7^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

25, 325, 703, 817, 1825, 2101, 2353, 2465, 3277, 4525, 6697, 8321, 10225, 11041, 11521, 12025, 13665, 14089, 19345, 20197, 20417, 20425, 25829, 29857, 29891, 35425, 38081, 39331, 46657, 49241, 49321, 50881, 58825, 64681, 75241, 75361, 76627, 78937, 79381

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..61 from Daniel Lignon)

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), this sequence (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[25000] + 1, eulerPseudoQ[#, 7] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(7, (2*n+1))^n == 1 ||  Mod(7, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262055 Euler pseudoprimes to base 8: composite integers such that abs(8^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

9, 21, 65, 105, 133, 273, 341, 481, 511, 561, 585, 1001, 1105, 1281, 1417, 1541, 1661, 1729, 1905, 2047, 2465, 2501, 3201, 3277, 3641, 4033, 4097, 4641, 4681, 4921, 5461, 6305, 6533, 6601, 7161, 8321, 8481, 9265, 9709, 10261, 10585, 10745, 11041, 12545

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..182 from Daniel Lignon)

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), A262054 (base 7), this sequence (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[11000] + 1, eulerPseudoQ[#, 8] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(8, (2*n+1))^n == 1 ||  Mod(8, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1541, 1559

Offset: 1

Reikku Kulon, Jun 21 2009

Near superset of the Sophie Germain primes (A005384), excluding 2 and 3: 2n + 1 is prime. Nearly all members of this sequence are also prime, but four members less than 10000 are composite: 1541 = 23 * 67, 2465 = 5 * 17 * 29, 3281 = 17 * 193, and 4961 = 11^2 * 41.
The congruence of n modulo 4 is evenly distributed between 1 and 3. n is congruent to 5 (mod 6) for all n less than two billion.
This sequence has roughly twice the density of the sequence (A158034) corresponding to the Diophantine equation
f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)),
and contains most members of that sequence. Those it does not contain are composite and often congruent to 3 (mod 6).
Composite terms appear to predominantly belong to A262051. - Bill McEachen, Aug 29 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A161897 A000040, A002515, A005384, A158034, A158035, A158036, A145918, A002943.

a161896 n = a161896_list !! (n-1)
a161896_list = [x | x <- [1..],
                    (9^x - 3*3^x - 4*x) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

is(n)=my(m=2*n*(2*n+1),t=Mod(3,m)^n); t^2-3*t==4*n \\ Charles R Greathouse IV, Nov 25 2014

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

cp's OEIS Frontend

A262053 Euler pseudoprimes to base 6: composite integers such that abs(6^((n - 1)/2)) == 1 mod n.

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A262054 Euler pseudoprimes to base 7: composite integers such that abs(7^((n - 1)/2)) == 1 mod n.

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A262055 Euler pseudoprimes to base 8: composite integers such that abs(8^((n - 1)/2)) == 1 mod n.

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A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

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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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