A158358 -id:A158358 - cp's OEIS frontend

A244065 Pseudoprimes to base 3 that are not squarefree.

121, 3751, 4961, 7381, 11011, 29161, 32791, 142901, 228811, 239701, 341341, 551881, 566401, 595441, 671671, 784201, 856801, 1016521, 1237951, 1335961, 1433971, 1804231

Offset: 1

Felix Fröhlich, Jun 19 2014

nonn

Must be divisible by the square of a Mirimanoff prime, A014127. - Charles R Greathouse IV, Jun 21 2014

Felix Fröhlich and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 196 terms from Fröhlich)

Cf. A005935, A158358, A014127.

for(n=2, 10^9, if(!isprime(n) && Mod(3, n)^(n-1)==1 && !issquarefree(n), print1(n, ", ")))

list(lim)=my(M=[11,1006003],v=List(),p2);for(i=1,#M,p2=M[i]^2;forstep(n=p2,lim,p2,if(Mod(3,n)^(n-1)==1,listput(v,n))));Set(v) \\ Good for lim <= 9.4 * 10^29; Charles R Greathouse IV, Jun 21 2014

A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

Original entry on oeis.org

12327121, 129816911251, 259621495381, 346157884801, 605767053061, 6317168754781, 6922923480721, 12634325182441, 18518799663001, 21634109682121, 24273469559431, 57114029344321, 65681131896901, 102718706568661, 135083316211741, 135818875521811, 153342494379361

Offset: 1

Felix Fröhlich, Sep 24 2014

nonn

Numbers k such that 2^k == 2 (mod k) and k is divisible by 3511^2.

Unless there are other Wieferich primes (A001220) besides 1093 and 3511, the intersection and the union of this sequence with A247830 are given by A219346 and A158358, respectively, and the even terms are given by A295740. - Max Alekseyev, Nov 26 2017 [The indices of the even terms in this sequence are 430, 525, 543, 701, 811, 826, 937, 1235, 1277, 1388, ... - Jianing Song, Feb 08 2019]

Jianing Song, Table of n, a(n) for n = 1..1470
R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473.
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.

Subsequence of each of (A001567 U A006935), A015919, A158358 composed of the terms divisible by 3511^2.

Cf. A001220, A013929, A219346, A247830, A295740.

vi=readvec("b158358.txt")
for(n=1, #vi, if(Mod(vi[n], 3511^2)==0, print1(vi[n], ", ")))

list(N)=select(k->Mod(2, k)^k==2, 3511^2*vector(N\3511^2\2, i, i)) \\ Jianing Song, Feb 07 2019

Name changed by Jianing Song, Feb 07 2019 to include the even pseudoprimes to base 2 (A006935) at the suggestion of Max Alekseyev.

A295740 Even pseudoprimes (A006935) that are not squarefree.

Original entry on oeis.org

190213279479817426, 283959621257123566, 301971651496560046, 575203724324614126, 800951203404568126, 849341919686285026, 1118572636403947726, 2080713636347910526, 2270517620327541586, 2767984602684877486, 5013069719001987826, 5133266340887464066, 5252931629341901506, 5743747078662858526

Offset: 1

Max Alekseyev, Nov 26 2017

nonn

For a prime p, if p^2 divides an even pseudoprime, then p is a Wieferich prime (A001220) and A007733(p)=A002326((p-1)/2) is odd. Currently, the only known such prime is p=3511.

So, all known terms are multiples of 3511^2. Furthermore, no term can be a multiple of 3511^3.

			a(1) = 190213279479817426 = 2 * 7 * 79 * 1951 * 3511^2 * 7151.
a(2) = 283959621257123566 = 2 * 599 * 937 * 3511^2 * 20521.
a(3) = 301971651496560046 = 2 * 31 * 71 * 73 * 3511^2 * 76231.

Max Alekseyev, Table of n, a(n) for n = 1..80 (5 wrong terms were removed by _Mauro Fiorentini_, May 31 2020)

Intersection of A006935 and A013929.

The even terms of A158358. Also, unless there is a Wieferich prime greater than 3511, the even terms of A247831.

A219346 Pseudoprimes to base 2 that are divisible by the squares of at least two primes.

Original entry on oeis.org

4578627124156945861, 57406119388190085241, 71879404939979986441, 98654983210791303661, 109509947374633729561, 153653468307592928221, 213717603347521018201, 541537521095562280381, 608114634633795825901, 721729926215346550321, 1062575800959998723581

Offset: 1

Charles R Greathouse IV, Nov 18 2012

nonn

2-pseudoprimes of the form n*p^2*q^2, where p and q are distinct Wieferich primes (A001220).

Unless there are other Wieferich primes besides 1093 and 3511, the sequence is the intersection of A247830 and A247831. - Max Alekseyev, Nov 26 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of each of the following sequences: A158358, A247830, A247831.

forstep(n=236641458619975501,4.489e31,723664277589495060,if(Mod(2,n)^(n-1)==1, print1(n", ")))

A243010 Pseudoprimes to base 5 that are not squarefree.

Original entry on oeis.org

4, 124, 11476, 59356, 80476, 91636, 250876, 261964, 482516, 1385836, 1926676, 2428084, 2589796, 3743476, 4101796, 6797764, 9155476, 10701076, 10743436, 11263396, 13799836, 13859956, 15570556, 20396476

Offset: 1

Felix Fröhlich, Aug 18 2014

nonn

Any term is divisible by the square of a base 5 Wieferich prime (A123692).

Intersection of A005936 and A013929. - Michel Marcus, Aug 21 2014

Amiram Eldar, Table of n, a(n) for n = 1..879 (terms below 10^12)

Cf. A005936, A123692, A158358, A243089, A243090, A244065.

forcomposite(n=1, 1e9, if(Mod(5, n)^(n-1)==1, if(!issquarefree(n), print1(n, ", "))))

A243089 Pseudoprimes to base 7 that are not squarefree.

Original entry on oeis.org

25, 325, 1825, 4525, 4825, 10225, 12025, 16725, 20425, 30025, 35425, 58825, 177025, 216525, 265525, 352225, 526825, 611425, 675925, 710425, 717025, 746425, 772525, 784225, 834025, 877825, 1125825, 1126225, 1439425, 1491025, 1579225, 1935025, 1973425, 2176525

Offset: 1

Felix Fröhlich, Aug 18 2014

nonn

Any term is divisible by the square of a base 7 Wieferich prime (A123693).

Intersection of A005938 and A013929. - Michel Marcus, Aug 21 2014

Amiram Eldar, Table of n, a(n) for n = 1..3095 (terms below 10^12)

Cf. A005938, A123693, A158358, A243010, A243090, A244065.

forcomposite(n=1, 1e9, if(Mod(7, n)^(n-1)==1, if(!issquarefree(n), print1(n, ", "))))

A243090 Pseudoprimes to base 8 that are not squarefree.

Original entry on oeis.org

9, 45, 63, 117, 153, 585, 2169, 4005, 9945, 13833, 17865, 27261, 33201, 36873, 40833, 57681, 69345, 69921, 95085, 140985, 155961, 161721, 171405, 186201, 189441, 192465, 203841, 240471, 242451, 244413, 316881, 321201, 406341, 481041, 482769, 488709, 501921

Offset: 1

Felix Fröhlich, Aug 18 2014

nonn

Any member of the sequence is divisible by the square of a base 8 Wieferich prime, of which only three cases are known, namely 3, 1093 and 3511.

Intersection of A020137 and A013929. - Michel Marcus, Aug 21 2014

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

Cf. A020137, A158358, A243010, A243089, A244065.

forcomposite(n=1, 1e9, if(Mod(8, n)^(n-1)==1, if(!issquarefree(n), print1(n, ", "))))

A247830 Pseudoprimes to base 2 divisible by 1093^2.

Original entry on oeis.org

1194649, 3914864773, 5654273717, 6523978189, 22178658685, 26092328809, 31310555641, 41747009305, 53053167441, 58706246509, 74795779241, 85667085141, 237865367741, 333967711897, 467032496113, 575310702877, 601401837037, 962329192917, 1087566636885

Offset: 1

Felix Fröhlich, Sep 24 2014

nonn

Numbers k such that 2^k == 2 (mod k) and k is divisible by 1093^2. - Jianing Song, Apr 07 2019
In contrast to A247831, all terms of this sequence are odd. - Max Alekseyev, Nov 26 2017
Unless there are other Wieferich primes (A001220) besides 1093 and 3511, the intersection and the union of this sequence with A247831 are given by A219346 and A158358, respectively. - Max Alekseyev, Nov 26 2017

R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473.
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.

Subsequence of each of A001567, A015919, A158358 composed of the terms divisible by 1093^2.

Cf. A001220, A013929, A219346, A247831.

vi=readvec("b158358.txt")
for(n=1, #vi, if(Mod(vi[n], 1093^2)==0, print1(vi[n], ", ")))

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, ", ")))

A306449 Pseudoprimes to base 10 that are not squarefree.

Original entry on oeis.org

9, 99, 657, 909, 1233, 11169, 13833, 19503, 20961, 23661, 51291, 69921, 90009, 99297, 109737, 139329, 203841, 237169, 256059, 321201, 339021, 346473, 460251, 475641, 686169, 760761, 927081, 1080801, 1621089, 1679931, 3100833, 3316941, 3845601, 3846051, 3942657, 4095081, 4281057

Offset: 1

Jianing Song, Feb 16 2019

nonn

Numbers k that are not squarefree and satisfy 10^(k-1) == 1 (mod k).
Any term is divisible by the square of a base-10 Wieferich prime (A045616 = {3, 487, 56598313, ...}).
Intersection of A005939 and A013929.

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

Pseudoprimes to base b that are not squarefree: A158358 (b=2), A244065 (b=3), A243010 (b=5), A243089 (b=7), A243090 (b=8), A306448 (b=9), this sequence (b=10).

Cf. also A045616, A005939, A013929.

for(n=1, 10^6, if(Mod(10, n)^(n-1)==1 && !issquarefree(n), print1(n, ", ")))

cp's OEIS Frontend

A244065 Pseudoprimes to base 3 that are not squarefree.

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A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

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A295740 Even pseudoprimes (A006935) that are not squarefree.

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A219346 Pseudoprimes to base 2 that are divisible by the squares of at least two primes.

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A243010 Pseudoprimes to base 5 that are not squarefree.

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A243089 Pseudoprimes to base 7 that are not squarefree.

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A243090 Pseudoprimes to base 8 that are not squarefree.

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A247830 Pseudoprimes to base 2 divisible by 1093^2.

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A306448 Pseudoprimes to base 9 that are not squarefree.

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