A337818 -id:A337818 - cp's OEIS frontend

A337846 Odd integers k such that 2^((k-1)/2) == 1 (mod k*(k-2)).

17, 257, 457, 1297, 6481, 11953, 26321, 47521, 47881, 49681, 65537, 74449, 157081, 165601, 278497, 333433, 476737, 557041, 560737, 576721, 1033057, 1266841, 1329337, 1463617, 1468897, 2291041, 2422201, 2754481, 2851633, 2969137, 3255281

Offset: 1

Benoit Cloitre, Sep 26 2020

nonn

Computed terms are prime. Is this a possible primality test or are there pseudo primes? Terms are of the form 8k+1.

The Fermat number F(5) = A000215(5) = 4294967297 = 641*6700417 is the smallest composite counterexample. - Hugo Pfoertner, Sep 26 2020

Cf. A081762, A337818.

Select[Range[3, 10^6, 2], PowerMod[2, (# - 1)/2, #*(# - 2)] == 1 &] (* Amiram Eldar, Sep 26 2020 *)

is(n) = n%2 && n>=3 && Mod(2, n*(n-2))^((n-1)/2) == 1

A337829 Odd integers k such that 5^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

Original entry on oeis.org

3, 7, 43, 53, 127, 163, 487, 677, 883, 2647, 8527, 8803, 14407, 18523, 26407, 32887, 35323, 39367, 71443, 105967, 184087, 184843, 230203, 265483, 319327, 388963, 425083, 543607, 554527, 651043, 688087, 690607, 698923, 796447, 887923, 924043, 1001323

Offset: 1

Benoit Cloitre, Sep 24 2020

nonn

The first 564 terms are prime.

Chai Wah Wu, Table of n, a(n) for n = 1..564

Cf. A337818.

Select[Range[3, 10^6, 2], PowerMod[5, (# - 1)/2, (t = #*(# - 1)/2)] == t - 1 &] (* Amiram Eldar, Sep 24 2020 *)

A337848 Odd integers k>=5 such that 2^((k-1)/2)-1 == 0 (mod k*(k-3)/2).

Original entry on oeis.org

73, 241, 2593, 5113, 8713, 18433, 53593, 55681, 86113, 102241, 126337, 127873, 158113, 181721, 184369, 186049, 208393, 219313, 221537, 241921, 262657, 267913, 282313, 314161, 314401, 341641, 362521, 398441, 415873, 450913, 534241, 619921, 651169, 731881, 953473, 1045801, 1153441, 1294177, 1554281, 2023921, 2162401, 2345401, 2533681

Offset: 1

Benoit Cloitre, Sep 26 2020

nonn

Computed terms are prime. Is it always the case? If not it would be interesting to compute the pseudoprimes.
1234125721 = 24841*49681, 4294901761 = 193*22253377, 6602556241 = 57457*114913 are composite counterexamples to the assumption that all terms are prime. - Hugo Pfoertner, Sep 26 2020
These are a(420), a(705) and a(830). Together with a(956) = 10025492401 = 101 * 701 * 141601 they are the first 4 composite terms. - Amiram Eldar, Jun 17 2022

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A337818.

Select[Range[5, 10^6, 2], PowerMod[2, (# - 1)/2, #*(# - 3)/2] == 1 &] (* Amiram Eldar, Sep 26 2020 *)

is(n) = n%2 && n>=3 && Mod(2, n*(n-3)/2)^((n-1)/2) ==1

A337858 Integers k>=3 such that 2^k == 2 (mod k(k-1)(k-2)/6).

Original entry on oeis.org

3, 5, 37, 101, 44101, 3766141, 8122501, 18671941, 35772661, 36969661, 208168381, 425420101, 725862061, 778003381, 818423101, 1269342901, 9049716901, 27221068981, 60138957061, 125980182901, 137330493301, 314912454781, 315322826869, 478543291381, 667933881301

Offset: 1

Benoit Cloitre, Sep 26 2020

nonn

Computed terms are prime. Is it always the case? If not it would be interesting to compute the smallest pseudoprime.
It seems that all larger terms are of the form 180*k + 1, starting at a(5) = 44101 = 180*245 + 1. - Hugo Pfoertner, Sep 27 2020
Other terms of the form 180*k+1 (which are all prime): 60138957061, 125980182901, 137330493301, 478543291381, 667933881301, 700212813301, 701030830501, 720023604301, 766514618101, 778382658901. - Chai Wah Wu, Oct 06 2020

Delbert L. Johnson, Table of n, a(n) for n = 1..28

Cf. A337818, A337846.

Select[Range[3, 10^7], PowerMod[2, #, #*(# - 1)*(# - 2)/6] == 2 &] (* Amiram Eldar, Sep 27 2020 *)

is(n) = n>=3 && Mod(2, n*(n-1)*(n-2)/6)^n ==2

a(12)-a(18) from Amiram Eldar, Sep 27 2020

a(19)-a(25) from Delbert L. Johnson, Mar 27 2024

A337828 Odd integers k such that 3^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

Original entry on oeis.org

5, 101, 62501, 1020101, 31562501, 139476101, 637562501, 789062501, 985502501, 2656262501, 7455062501, 19726562501, 53662562501

Offset: 1

Benoit Cloitre, Sep 24 2020

Computed terms are prime.

Cf. A337818.

Select[Range[3, 3*10^7, 2], PowerMod[3, (# - 1)/2, (t = #*(# - 1)/2)] == t - 1 &] (* Amiram Eldar, Sep 24 2020 *)

a(7)-a(13) from Amiram Eldar, Sep 25 2020

A337830 Odd integers k such that 6^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

Original entry on oeis.org

2843, 2390123, 9893003, 16236347, 46353707, 334358459, 564092747, 584214107, 1640200619, 2010092603, 14044030043, 22315857803, 23753097803, 92758244699, 136542051227, 281195463179, 332945964107, 545960571227

Offset: 1

Benoit Cloitre, Sep 24 2020

Computed terms are prime.

Conjecture: a(n) == 1 mod 406 for n > 5. - Chai Wah Wu, Oct 07 2020

Cf. A337818.

Select[Range[3, 10^7, 2], PowerMod[6, (# - 1)/2, (t = #*(# - 1)/2)] == t - 1 &] (* Amiram Eldar, Sep 24 2020 *)

a(6)-a(13) from Amiram Eldar, Sep 24 2020
a(14)-a(15) from Bill McEachen, Jul 21 2025
a(16)-a(18) from Bill McEachen, Aug 03 2025

A337831 Odd integers k such that 7^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

Original entry on oeis.org

5, 101, 62501, 1020101, 3512501, 12650501, 22598021, 31562501, 365328101, 789062501, 11412000101, 64014060101

Offset: 1

Benoit Cloitre, Sep 24 2020

Computed terms are prime.

Cf. A337818.

Select[Range[3, 10^7, 2], PowerMod[7, (# - 1)/2, (t = #*(# - 1)/2)] == t - 1 &] (* Amiram Eldar, Sep 24 2020 *)

a(9)-a(12) from Amiram Eldar, Sep 25 2020

A337847 Odd integers k such that 3^((k-1)/2) == 1 (mod k*(k-2)).

Original entry on oeis.org

457, 1297, 6481, 14401, 26497, 44101, 47521, 47881, 165601, 225457, 446881, 560737, 576721, 677041, 1037857, 1049941, 1649341, 1903981, 1934137, 2291041, 3990601, 4110121, 4262161, 4663297, 4736341, 5293081, 5317057, 5372929, 6410497, 6535681, 6651361, 8122501

Offset: 1

Benoit Cloitre, Sep 26 2020

nonn

Computed terms are prime. Is this a possible primality test or are there pseudo primes? Terms are of the form 12k+1.

Cf. A081762, A337818.

Select[Range[3, 10^6, 2], PowerMod[3, (# - 1)/2, #*(# - 2)] == 1 &] (* Amiram Eldar, Sep 26 2020 *)

is(n) = n%2 && n>=3 && Mod(3, n*(n-2))^((n-1)/2) == 1

More terms from Amiram Eldar, Sep 26 2020

A337859 k-1 for integers k>=4 such that 2^k == 4 (mod k(k-1)(k-2)*(k-3)/24).

Original entry on oeis.org

3, 5, 37, 44101, 157081, 2031121, 7282801, 8122501, 18671941, 78550201, 208168381, 770810041, 2658625201, 2710529641, 5241663001, 14643783001, 18719308441, 56181482281, 73303609681, 74623302001, 110102454001, 140659081201

Offset: 1

Benoit Cloitre, Sep 26 2020

Computed terms are prime. Is it always the case? Probably not and it would be interesting to compute the smallest pseudoprime.

It seems that all larger terms are of the form 60*k + 1, starting at a(4) = 44101 = 60*735 + 1. Further terms of this form after a(17) are 56181482281, 73303609681, 74623302001, 110102454001, 140659081201, 283268822761, 469078212241, 530106748081, 570417709681, 701030830501, 720023604301; all are prime. - Hugo Pfoertner, Sep 28 2020

Cf. A337818, A337846.

Select[Range[4, 10^7], (t = #*(# - 1)*(# - 2)*(# - 3)/24) == 1 || PowerMod[2, #, t] == 4 &] - 1 (* Amiram Eldar, Sep 27 2020 *)

is(k) = k>=4 && Mod(2,k*(k-1)*(k-2)*(k-3)/24)^k == 4

a(13)-a(17) from Amiram Eldar, Sep 27 2020

a(18)-a(22) from Chai Wah Wu, Oct 09 2020

cp's OEIS Frontend

A337846 Odd integers k such that 2^((k-1)/2) == 1 (mod k*(k-2)).

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A337829 Odd integers k such that 5^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

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A337848 Odd integers k>=5 such that 2^((k-1)/2)-1 == 0 (mod k*(k-3)/2).

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A337858 Integers k>=3 such that 2^k == 2 (mod k*(k-1)*(k-2)/6).

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A337828 Odd integers k such that 3^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

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A337830 Odd integers k such that 6^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

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A337831 Odd integers k such that 7^((k-1)/2) + 1 == 0 (mod k*(k-1)/2).

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A337847 Odd integers k such that 3^((k-1)/2) == 1 (mod k*(k-2)).

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Extensions

A337859 k-1 for integers k>=4 such that 2^k == 4 (mod k*(k-1)*(k-2)*(k-3)/24).

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A337858 Integers k>=3 such that 2^k == 2 (mod k(k-1)(k-2)/6).

A337859 k-1 for integers k>=4 such that 2^k == 4 (mod k(k-1)(k-2)*(k-3)/24).