A005935 -id:A005935 - cp's OEIS frontend

A083734 Pseudoprimes to bases 3 and 5.

1541, 1729, 1891, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 88831, 101101, 112141, 115921, 126217, 146611, 162401, 172081, 188461, 218791, 252601, 294409, 314821, 334153, 340561, 342271, 399001, 410041, 416641

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

			a(1)=1541 since it is the first nonprime number such that 3^(k-1) = 1 (mod k) and 5^(k-1) = 1 (mod k). - clarified by _Harvey P. Dale_, Jan 29 2013

Amiram Eldar, Table of n, a(n) for n = 1..15806 (terms 1..147 from R. J. Mathar)
F. Richman, Primality testing with Fermat's little theorem

Intersection of A005935 and A005936.

Select[Range[420000],!PrimeQ[#]&&PowerMod[3,#-1,#]==PowerMod[5,#-1,#]==1&] (* Harvey P. Dale, Jan 29 2013 *)

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k) and 5^(k-1) = 1 (mod k).

A217841 Fermat pseudoprimes n to base 3 for which sqrt(8*n + 1) is an integer.

Original entry on oeis.org

91, 703, 1891, 2701, 7381, 8911, 10585, 12403, 16471, 18721, 29161, 38503, 41041, 49141, 79003, 88831, 93961, 104653, 115921, 146611, 188191, 218791, 226801, 269011, 286903, 314821, 334153, 364231, 385003, 497503, 534061, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1237951

Offset: 1

Marius Coman, Oct 12 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Cf. A005935, A210461 (subsequence).

list(lim)=my(v=List(),n); lim\=1; forstep(k=27,sqrtint(8*lim+1),2, n=k^2>>3; if(Mod(3,n)^(n-1)==1, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

a(15)-a(18) and a(35) from Charles R Greathouse IV, Jun 30 2017

A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

Original entry on oeis.org

91, 7381, 597871, 48427561, 3922632451, 317733228541, 25736391511831, 2084647712458321, 168856464709124011, 1107867264956562636991, 588766087155780604365200461, 47690053059618228953581237351, 25344449488056571213320166359119221, 166284933091139163730593611482181209801

Offset: 1

Marius Coman, Oct 12 2012

nonn

These numbers were obtained for values of k from 1 to 20, with the following exceptions: k = 10, 12, 13, 16, 17, 19, for which were obtained 3^n mod n = 3^7, 3^31, 3^37, 3^25, 3^31, 3^13.
Conjecture: There are infinitely many Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8, where k is a natural number.
It is true: for example, when 2k+1 is a prime number (see A210461). - Bruno Berselli, Jan 22 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..190
Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Cf. A005935, A210461 (subsequence), A217841.

Select[Table[(3^(4k + 2) - 1)/8, {k, 80}], PowerMod[3, # - 1, #] == 1 &] (* Alonso del Arte, May 14 2019 *)

list(lim)=my(v=List(),t); lim\=1; for(k=1,(logint(8*lim+1,3)-2)\4, t=3^(4*k + 2)>>3; if(Mod(3,t)^t==3, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

A306451 Non-coprime pseudoprimes or primes to base 3: numbers k that are multiples of 3 and are such that k divides 3^k - 3.

Original entry on oeis.org

3, 6, 66, 561, 726, 7107, 8205, 8646, 62745, 100101, 140097, 166521, 237381, 237945, 566805, 656601, 876129, 1053426, 1095186, 1194285, 1234806, 1590513, 1598871, 1938021, 2381259, 2518041, 3426081, 4125441, 5398401, 5454681, 5489121, 5720331, 5961441

Offset: 1

Jianing Song, Feb 17 2019

nonn

Union of {3} and (A122780 - {1} - A005935).
Numbers of the form 3*m such that 3^(3*m-1) == 1 (mod m).
The squarefree terms are listed in A306450.

Jianing Song, Table of n, a(n) for n = 1..163 (all terms below 10^9)

Cf. A005935, A006935, A122780, A306450.

A258801 is a subsequence.

forstep(n=3, 1e7, 3, if(Mod(3, n)^n==3, print1(n, ", ")))

66 is a term because 66 divides 3^66 - 3 = 3*(3^65 - 1) = 3*(3^5 - 1)*(3^60 + 3^55 + ... + 3^5 + 1) and 66 is divisible by 3.

A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

Original entry on oeis.org

2701, 18721, 31621, 49141, 83333, 90751, 104653, 226801, 282133, 653333, 665281, 721801, 873181, 1373653, 1530787, 1537381, 1584133, 1690501, 1755001, 1987021, 2008597, 2035153, 2284453, 2746589, 2944261, 3059101, 3116107, 3363121, 3375041, 3375487, 4082653, 4314967

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

The first term that has more than 2 prime factors is a(1067) = A333131(1) = 11500521553.

The first term that is also a Carmichael number is a(1131) = 13079177569.

			2701 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are all primes: 37 and 73.

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

Intersection of A050217 and A328662.

Subsequence of A001567, A005935, A052155 and A153513.

pspQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; Select[Range[10^6], pspQ]

A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

Original entry on oeis.org

11500521553, 13079177569, 52474339009, 168003672409, 229352039821, 280792563977, 318289021201, 428178002569, 918660756421, 2015841188197, 2367478228501, 2544457029601, 2639665216117, 3023595814801, 3457449931321, 3712164285421, 4348114583017, 6046196043229

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

Up to 2^64 all the 1085 terms are nonsquarefree, 2 terms have 4 prime factors: a(163) = 18362297383286473 = 3037 * 6073 * 9109 * 109297 and a(651) = 2587580959818925201 = 18121 * 36241 * 54361 * 72481, and no term have more than 4 prime factors.

			11500521553 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are either primes (937, 1873, 6553) or Fermat pseudoprimes to both bases 2 and 3 (1755001, 6140161, 12273769, 11500521553).

Amiram Eldar, Table of n, a(n) for n = 1..1085 (terms below 2^64)

Subsequence of A001567, A005935, A050217, A052155, A153513, A178997, A328662, A328663, A333130.

pspQ[n_] := PrimeOmega[n] > 2 && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; seq = {}; Do[If[pspQ[n], AppendTo[seq, n]], {n, 1, 6*10^10}]; seq

A057943 Numbers k such that the smallest palindromic pseudoprime to base k is 101101.

Original entry on oeis.org

2, 51, 60, 75, 96, 200, 207, 279, 288, 348, 402, 432, 464, 492, 500, 531, 555, 590, 646, 652, 662, 675, 695, 732, 750, 790, 843, 855, 860, 888, 894, 920, 927, 983, 984, 1074, 1102, 1139, 1140, 1150, 1152, 1163, 1164, 1203, 1215, 1230, 1251, 1278, 1283, 1336

Offset: 1

Patrick De Geest, Oct 15 2000

			2 is a term since 101101 is the least Fermat pseudoprime to base 2 (A001567) which is also a palindrome in base 10 (A002113).
3 is not a term since the least Fermat pseudoprime to base 3 (A005935) which is also a palindrome in base 10 is 121.

Amiram Eldar, Table of n, a(n) for n = 1..10000
C. K. Caldwell, Carmichael number
C. K. Caldwell, Pseudoprimes
C. Rivera, Carmichael Numbers
Index entries for sequences related to pseudoprimes

Cf. A001567, A002113, A002997.

palinComps = Select[Range[2, 101100], PalindromeQ[#] && CompositeQ[#] &]; seqQ[n_] := PowerMod[n, 101100, 101101] == 1 && AllTrue[palinComps, PowerMod[n, #-1, #] != 1 &]; Select[Range[1336], seqQ] (* Amiram Eldar, Jan 30 2020 *)

A083738 Pseudoprimes to bases 2,3 and 7.

Original entry on oeis.org

1105, 2465, 10585, 18721, 29341, 46657, 75361, 104653, 115921, 162401, 226801, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 534061, 552721, 574561, 658801, 721801, 852841, 1024651

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

			a(1)=1105 since it is the first number such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Amiram Eldar, Table of n, a(n) for n = 1..10460 (terms 1..98 from R. J. Mathar)
F. Richman, Primality testing with Fermat's little theorem

Intersection of A001567 and A083735. Intersection of A005935 and A083733. - R. J. Mathar, Apr 05 2011

Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[2, #-1,#] == PowerMod[3, #-1,#] == PowerMod[7, #-1,#] == 1&] (* Amiram Eldar, Jun 29 2019 *)

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A083740 Pseudoprimes to bases 3,5 and 7.

Original entry on oeis.org

29341, 46657, 75361, 88831, 115921, 146611, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 721801, 852841, 954271, 1024651, 1152271, 1193221, 1314631, 1461241, 1569457, 1615681

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

			a(1)=29341 since it is the first number such that 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Amiram Eldar, Table of n, a(n) for n = 1..8702 (terms 1..77 from R. J. Mathar)
F. Richman, Primality testing with Fermat's little theorem

Cf. A005935, A005936, A005938, A083734, A083735.

Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[3, #-1, #] == PowerMod[5, #-1, #] == PowerMod[7, #-1, #] == 1&]

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Intersection of A083734 and A005938. Intersection of A083735 and A005936. - R. J. Mathar, Apr 05 2011

A247033 Numbers of the form (3^k - 3)/k.

Original entry on oeis.org

0, 3, 8, 48, 121, 312, 16104, 122640, 7596480, 61171656, 4093181688, 2366564736720, 19924948267224, 12169835294351280, 889585277491970400, 7633882962663652968, 565719454445904325272, 365721616371321130128240, 239498069351503974657030696, 2084811062715550992506283600

Offset: 1

Juri-Stepan Gerasimov, Sep 09 2014

nonn

Generated by k: 1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 66, 67, ...

Subsequence of A246445.

			121 is in this sequence because (3^k - 3)/k = (3^6 - 3)/6 = 121.

Cf. A000040, A005935, A064535 (with form (2^k - 2)/k), A122780 (nonprimes k in a(n)), A246445, A247307 (with form (4^k - 4)/k).

cp's OEIS Frontend

A083734 Pseudoprimes to bases 3 and 5.

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A217841 Fermat pseudoprimes n to base 3 for which sqrt(8*n + 1) is an integer.

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A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

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A306451 Non-coprime pseudoprimes or primes to base 3: numbers k that are multiples of 3 and are such that k divides 3^k - 3.

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A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

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A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

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A057943 Numbers k such that the smallest palindromic pseudoprime to base k is 101101.

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A083738 Pseudoprimes to bases 2,3 and 7.

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A083740 Pseudoprimes to bases 3,5 and 7.