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.

A344828 a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2) and prime(n+3) are all base-b Wieferich primes.

Original entry on oeis.org

557, 1207, 1451, 13543, 24675, 39016, 217682, 165407, 1357748, 399254, 1146590, 325346, 1895206, 3365181, 4674177, 21251205, 40698745, 6795147, 36463448, 12717474, 54383927, 7411274, 35989426, 101112784, 86045167, 13128506, 276293632, 169093089, 223680564, 137073637
Offset: 1

Views

Author

Felix Fröhlich, May 29 2021

Keywords

Crossrefs

Cf. A039678, A259075. Column 4 of A286816.
Cf. smallest b > 1 such that prime(n+i) is a base-b Wieferich prime for each i = 0..k: A039678 (k=0), A259075 (k=1), A344827 (k=2), A344829 (k=4), A344830 (k=5), A344831 (k=6), A344832 (k=7).

Programs

  • PARI
    a(n) = my(v=[prime(n)]); while(#v < 4, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

Extensions

Terms a(24) onward from Max Alekseyev, Oct 10 2023

A344829 a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2), prime(n+3) and prime(n+4) are all base-b Wieferich primes.

Original entry on oeis.org

19601, 54568, 13543, 296449, 3414284, 14380864, 3727271, 7916603, 65097619, 13793462, 152541840, 30495845, 91779237, 183068599, 558175167, 40698745, 825287029, 2151529020, 6271678163, 1266687934, 3149182509, 989067909, 10785363668, 18739432977, 4877709531, 24531035970, 11683733786, 52383593584
Offset: 1

Views

Author

Felix Fröhlich, May 29 2021

Keywords

Crossrefs

Cf. A039678, A259075. Column 5 of A286816.
Cf. smallest b > 1 such that prime(n+i) is a base-b Wieferich prime for each i = 0..k: A039678 (k=0), A259075 (k=1), A344827 (k=2), A344828 (k=3), A344830 (k=5), A344831 (k=6), A344832 (k=7).

Programs

  • PARI
    a(n) = my(v=[prime(n)]); while(#v < 5, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

Extensions

Terms a(22) onward from Max Alekseyev, Oct 10 2023

A344830 a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2), prime(n+3), prime(n+4) and prime(n+5) are all base-b Wieferich primes.

Original entry on oeis.org

132857, 2006776, 296449, 17134811, 36763941, 34998229, 31239565, 968576295, 1027038511, 287811239, 15022368222, 33452659960, 19477999997, 132892045949, 47341045210, 32176849766, 106967760951, 303459122992, 20216391690, 1411108384416, 219517083156, 361244580521, 455588981749
Offset: 1

Views

Author

Felix Fröhlich, May 29 2021

Keywords

Crossrefs

Cf. A039678, A259075. Column 6 of A286816.
Cf. smallest b > 1 such that prime(n+i) is a base-b Wieferich prime for each i = 0..k: A039678 (k=0), A259075 (k=1), A344827 (k=2), A344828 (k=3), A344829 (k=4), A344831 (k=6), A344832 (k=7).

Programs

  • PARI
    a(n) = my(v=[prime(n)]); while(#v < 6, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

Extensions

Terms a(10) onward from Max Alekseyev, Oct 10 2023

A344831 a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2), prime(n+3), prime(n+4), prime(n+5) and prime(n+6) are all base-b Wieferich primes.

Original entry on oeis.org

4486949, 20950343, 23250274, 741652533, 710808570, 2380570527, 4967352848, 39489360341, 6143373023, 64470545267, 627069771908, 407165268367, 676579634125, 676579634125, 2923957743077, 369072416795, 5110842013966, 19645397150616, 16037329738682, 5867065121893, 31015370285102
Offset: 1

Views

Author

Felix Fröhlich, May 29 2021

Keywords

Comments

Surprisingly, we have a(13) = a(14) = A344832(13). - Max Alekseyev, Oct 10 2023

Crossrefs

Cf. A039678, A259075. Column 7 of A286816.
Cf. smallest b > 1 such that prime(n+i) is a base-b Wieferich prime for each i = 0..k: A039678 (k=0), A259075 (k=1), A344827 (k=2), A344828 (k=3), A344829 (k=4), A344830 (k=5), A344832 (k=7).

Programs

  • PARI
    a(n) = my(v=[prime(n)]); while(#v < 7, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

Extensions

Terms a(8) onward from Max Alekseyev, Oct 10 2023

A344832 a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2), prime(n+3), prime(n+4), prime(n+5), prime(n+6) and prime(n+7) are all base-b Wieferich primes.

Original entry on oeis.org

126664001, 230695118, 882345432, 12106746963, 93732236423, 66888229817, 84391291750, 3685694924698, 4506100208066, 1058174730735, 3827951700972, 58674393094169, 676579634125, 83450880181400, 100819901293157, 365919682326848, 695001153578920, 1046021079620904, 2989564836636529, 3724954519064878
Offset: 1

Views

Author

Felix Fröhlich, May 29 2021

Keywords

Crossrefs

Cf. A039678, A259075. Column 8 of A286816.
Cf. smallest b > 1 such that prime(n+i) is a base-b Wieferich prime for each i = 0..k: A039678 (k=0), A259075 (k=1), A344827 (k=2), A344828 (k=3), A344829 (k=4), A344830 (k=5), A344831 (k=6).

Programs

  • PARI
    a(n) = my(v=[prime(n)]); while(#v < 8, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

Extensions

Terms a(5) onward from Max Alekseyev, Oct 10 2023

A344827 a(n) is the smallest b > 1 such that prime(n), prime(n+1) and prime(n+2) are all base-b Wieferich primes.

Original entry on oeis.org

449, 226, 1207, 606, 3469, 653, 5649, 26645, 7805, 6154, 36088, 14368, 49662, 66565, 153463, 40667, 760637, 31871, 265418, 411467, 484205, 148989, 688285, 796095, 1920186, 747071, 3516680, 569812, 905979, 3193580, 3303343, 1967646, 1728157, 4436267, 912246
Offset: 1

Views

Author

Felix Fröhlich, May 29 2021

Keywords

Crossrefs

Cf. A039678, A259075. Column 3 of A286816.
Cf. smallest b > 1 such that prime(n+i) is a base-b Wieferich prime for each i = 0..k: A039678 (k=0), A259075 (k=1), A344828 (k=3), A344829 (k=4), A344830 (k=5), A344831 (k=6), A344832 (k=7).

Programs

  • PARI
    a(n) = my(v=[prime(n)]); while(#v < 3, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

A258787 Triangle read by rows: T(n, k) = smallest base b > 1 such that p = prime(n) is the k-th base-b Wieferich prime for k = 1, 2, 3, ..., n.

Original entry on oeis.org

5, 8, 17, 7, 26, 449, 30, 18, 197, 557, 3, 9, 118, 1207, 19601, 22, 146, 19, 361, 8201, 132857, 38, 40, 224, 249, 4625, 296449, 4486949, 54, 28, 68, 99, 4033, 4625, 296449, 126664001, 42, 130, 28, 118, 557, 8201, 997757, 24800401, 2363321449, 14, 41, 374, 1745, 901, 46826, 217682, 9312157, 758427193, 5229752849
Offset: 1

Views

Author

Felix Fröhlich, Jun 10 2015

Keywords

Examples

			T(4, 3) = 197, because 197 is the smallest base b such that p = prime(4) = 7 is the 3rd base-b Wieferich prime.
Triangle T(n, k) starts:
  5;
  8,  17;
  7,  26,  449;
  30, 18,  197, 557;
  3,  9,   118, 1207, 19601;
  22, 146, 19,  361,  8201,  132857;
  38, 40,  224, 249,  4625,  296449, 4486949;
  54, 28,  68,  99,   4033,  4625,   296449,  126664001;
  42, 130, 28,  118,  557,   8201,   997757,  24800401,  2363321449;
		

Crossrefs

Cf. A256236 (diagonal). A286816.

Programs

  • PARI
    nextwiefbase(n, a) = a++; while(Mod(a, n^2)^(n-1)!=1, a++); a
    wiefrank(n, a) = i=0; forprime(p=1, n, if(Mod(a, p^2)^(p-1)==1, i++)); i
    trianglerows(n) = i=1; while(i <= n, p=prime(i); for(k=1, i, b=2; while(wiefrank(p, b)!=k, b=nextwiefbase(p, b)); print1(b, ", ")); print(""); i++)
    trianglerows(9) \\ print first nine rows of the triangle

Extensions

More terms from Max Alekseyev, Oct 14 2023
Showing 1-7 of 7 results.