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 1-8 of 8 results.

A286816 Smallest b such that the k consecutive primes starting with prime(n) are all base-b Wieferich primes, i.e., satisfy b^(p-1) == 1 (mod p^2). Square array A(n, k), read by antidiagonals downwards.

Original entry on oeis.org

5, 17, 8, 449, 26, 7, 557, 226, 18, 18, 19601, 1207, 1207, 148, 3, 132857, 54568, 1451, 606, 239, 19, 4486949, 2006776, 13543, 13543, 3469, 249, 38, 126664001, 20950343, 296449, 296449, 24675, 653, 423, 28, 2363321449, 230695118, 23250274, 17134811, 3414284, 39016, 5649, 28, 28, 5229752849, 5229752849, 882345432, 741652533, 36763941, 14380864, 217682, 26645, 63, 14
Offset: 1

Views

Author

Felix Fröhlich, May 27 2017

Keywords

Examples

			The sequence of base-226 Wieferich primes starts 3, 5, 7, 97, 157, ... Since 226 is the smallest b such that the three consecutive primes starting with prime(2) = 3 are base-b Wieferich primes, A(2, 3) = 226.
Array starts:
n=1: 5, 17, 449, 557, 19601, 132857
n=2: 8, 26, 226, 1207, 54568, 2006776
n=3: 7, 18, 1207, 1451, 13543, 296449
n=4: 18, 148, 606, 13543, 296449, 17134811
n=5: 3, 239, 3469, 24675, 3414284, 36763941
n=6: 19, 249, 653, 39016, 14380864, 34998229

Crossrefs

Columns: A039678 (k=1), A259075 (k=2), A344827 (k=3), A344828 (k=4), A344829 (k=5), A344830 (k=6), A344831 (k=7), A344832 (k=8).
Cf. A256236 (row n=1), A258787.

Programs

  • PARI
    primevec(initialp, vecsize) = my(v=[initialp]); while(#v < vecsize, v=concat(v, nextprime(v[#v]+1))); v
a(n, k) = my(v=primevec(prime(n), k), b=2, i=0); while(1, for(x=1, #v, if(Mod(b, v[x]^2)^(v[x]-1)!=1, i++; break)); if(i==0, return(b)); b++; i=0)
array(rows, cols) = for(s=1, rows, for(t=1, cols, print1(a(s, t), ", ")); print(""))
array(5, 6) \\ print 5 X 6 array

Extensions

More terms from Max Alekseyev, Oct 10 2023

A319059 A(n, k) is the k-th number b > 1 such that b^(prime(n+i)-1) == 1 (mod prime(n+i)^2) for each i = 0..1, with k running over the positive integers; square array, read by antidiagonals, downwards.

Original entry on oeis.org

17, 37, 26, 53, 82, 18, 73, 107, 68, 148, 89, 118, 99, 215, 239, 109, 143, 226, 362, 360, 249, 125, 199, 276, 606, 485, 577, 423, 145, 224, 293, 717, 596, 653, 653, 28, 161, 226, 324, 753, 606, 868, 2098, 784, 63, 181, 251, 374, 766, 699, 1520, 2526, 1921, 571
Offset: 1

Views

Author

Felix Fröhlich, Sep 09 2018

Keywords

Examples

			The array starts as follows:
    17,   37,   53,    73,    89,   109,   125,   145,   161,   181,   197,   217
    26,   82,  107,   118,   143,   199,   224,   226,   251,   307,   332,   343
    18,   68,   99,   226,   276,   293,   324,   374,   393,   557,   607,   618
   148,  215,  362,   606,   717,   753,   766,  1207,  1304,  1322,  1371,  1451
   239,  360,  485,   596,   606,   699,   844,   846,   995,  1330,  1371,  1451
   249,  577,  653,   868,  1520,  1948,  1958,  2098,  2178,  2446,  2536,  2850
   423,  653, 2098,  2526,  2889,  3180,  4270,  4400,  4625,  4755,  5416,  5531
    28,  784, 1921,  2234,  2293,  3004,  4233,  4566,  4631,  4762,  4938,  5353
    63,  571, 1545,  3304,  3585,  3969,  4204,  5420,  6995,  7583,  7765,  7805
   374, 1492, 2509,  3323,  3405,  4472,  5651,  6154,  6492,  7805, 12348, 13040
   117, 1693, 2157,  4431,  4688,  6154,  6728,  6844,  6962,  9089, 11533, 13689
   787, 1368, 3214,  4106,  4895,  5552,  5830,  5900,  8892,  9229, 11389, 14272
  2059, 2152, 5548,  8354, 10557, 14368, 20320, 27657, 29296, 29945, 31434, 31452
  1085, 1771, 2210, 17902, 18793, 19679, 23670, 23676, 24298, 24928, 25885, 31800
   655, 1586, 1914,  3330,  3818,  7772,  8765,  9436,  9459, 12087, 13183, 24501

Crossrefs

Cf. A244249, A256236, A259075 (column 1).
Cf. analog for i = 0..t: A319060 (t=2), A319061 (t=3), A319062 (t=4), A319063 (t=5), A319064 (t=6), A319065 (t=7).

Programs

  • Mathematica
    rows = 10; t = 1;
T = Table[lst = {}; b = 2;
   While[Length[lst] < rows,
    p = Prime[n + Range[0, t]];
    If[AllTrue[PowerMod[b, (p - 1), p^2], # == 1 &], AppendTo[lst, b]]; b++];
   lst, {n, rows}];
T // TableForm (* Print the A(n,k) table *)
Flatten[Table[T[[j, i - j + 1]], {i, 1, rows}, {j, 1, i}]] (* Robert Price, Sep 30 2019 *)
  • PARI
    printrow(n, terms) = my(c=0); for(b=2, oo, my(j=0); for(i=0, 1, my(p=prime(n+i)); if(Mod(b, p^2)^(p-1)==1, j++)); if(j==2, print1(b, ", "); c++); if(c==terms, break))
array(rows, cols) = for(x=1, rows, printrow(x, cols); print(""))
array(8, 10) \\ print initial 8 rows and 10 columns of array

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

Links

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

Links

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

Links

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

Links

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

Links

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)))))
Showing 1-8 of 8 results.

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