cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Felix Fröhlich, Aug 18 2014

Keywords

Comments

Any term is divisible by the square of a base 5 Wieferich prime (A123692).
Intersection of A005936 and A013929. - Michel Marcus, Aug 21 2014

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

Felix Fröhlich, Aug 18 2014

Keywords

Comments

Any term is divisible by the square of a base 7 Wieferich prime (A123693).
Intersection of A005938 and A013929. - Michel Marcus, Aug 21 2014

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

Felix Fröhlich, Aug 18 2014

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Views

Author

Jianing Song, Feb 16 2019

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

  • PARI
    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

Views

Author

Jianing Song, Feb 16 2019

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

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

A306450 Non-coprime pseudoprimes to base 3 (A306451) that are not squarefree.

Original entry on oeis.org

726, 1053426, 6498426, 7912311, 8141001, 190381521, 202730781, 283975626, 524245326, 767159481, 1095790641, 1620456321, 1904467521, 2287621281, 2700546486, 3462782961, 4120800321, 4928482581, 5816852481, 5974336401, 9313587921, 18723332001, 21215225361, 22073079666, 29882080866, 30132305841
Offset: 1

Views

Author

Jianing Song, Feb 16 2019

Keywords

Comments

Intersection of A306451 and A013929.
Terms must be divisible by the square of a Mirimanoff prime p (or base-3 Wieferich prime, A014127) such that the multiplicative order of 3 modulo p is not divisible by 3. So far, the only known Mirimanoff primes are 11 and 1006003. The multiplicative order of 3 modulo 11 is 5, not a multiple of 3, while the multiplicative order of 3 modulo 1006003 is 1006002, which is a multiple of 3. As a result, all known terms are divisible by 3*11^2 = 363.

Examples

			726 is a term because 726 divides 3^726 - 3 and 726 = 2 * 3 * 11^2.

Crossrefs

Cf. A295740, A244065, A306451, A306452.

Programs

  • PARI
    forstep(n=3, 10^9, 3, if(Mod(3, n)^n==3 && !issquarefree(n), print1(n, ", ")))

Extensions

More terms from Jinyuan Wang, Feb 18 2019

A306452 Pseudoprimes to base 3 that are not squarefree, including the non-coprime pseudoprimes.

Original entry on oeis.org

121, 726, 3751, 4961, 7381, 11011, 29161, 32791, 142901, 228811, 239701, 341341, 551881, 566401, 595441, 671671, 784201, 856801, 1016521, 1053426, 1237951, 1335961, 1433971, 1804231, 1916761, 2000251, 2254351, 2446741, 2817001, 2983981, 3078361, 3307051, 3562361
Offset: 1

Views

Author

Jianing Song, Feb 17 2019

Keywords

Comments

Numbers k such that 3^k == 3 (mod k) and k is divisible by the square of a Mirimanoff prime (or base-3 Wieferich prime), A014127.
A non-coprime pseudoprime in base b is a number k such that b^k == b (mod k) and that gcd(b, k) > 1, and the non-coprime pseudoprime in base 3 (726, 1053426, 6498426, ...) that are not squarefree are listed in A306450 while the others terms in this sequence (121, 3751, 4961, ...) are listed in A244065. So this sequence is the union of A244065 and A306450.
Intersection of A122780 and A013929.

Examples

			121 is a term because 3^120 == (3^5)^24 == 1 (mod 121) and 121 = 11^2.
Although 3^725 = 243 rather than 1 mod 726, we see that nevertheless 3^726 = 3 mod 726, and since 726 = 2 * 3 * 11^2, 726 is in the sequence. - _Alonso del Arte_, Mar 16 2019

Links

Crossrefs

Cf. A013929, A014127, A122780, A158358, A244065, A306450.

Programs

  • Mathematica
    Select[Range[5000], PowerMod[3, #, #] == 3 && MoebiusMu[#] == 0 &] (* Alonso del Arte, Mar 16 2019 *)
  • PARI
    forcomposite(n=1, 10^7, if(Mod(3, n)^n==3 && !issquarefree(n), print1(n, ", ")))
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.