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 1-6 of 6 results.

A339531 Numbers b > 1 such that the smallest two primes, i.e., 2 and 3 are base-b Wieferich primes.

Original entry on oeis.org

17, 37, 53, 73, 89, 109, 125, 145, 161, 181, 197, 217, 233, 253, 269, 289, 305, 325, 341, 361, 377, 397, 413, 433, 449, 469, 485, 505, 521, 541, 557, 577, 593, 613, 629, 649, 665, 685, 701, 721, 737, 757, 773, 793, 809, 829, 845, 865, 881, 901, 917, 937, 953
Offset: 1

Views

Author

Felix Fröhlich, Dec 08 2020

Keywords

Links

Crossrefs

Cf. A256236, A263941. Row 1 of A319059.
Cf. smallest k primes are base-b Wieferich primes: A339532 (k=3), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Programs

  • Mathematica
    Select[Range[2, 10^3], Function[b, AllTrue[{2, 3}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)
  • PARI
    is(n) = forprime(p=1, 3, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

Formula

a(n) = 4*A263941(n) + 1 for n>=2, a(n) = 4*floor((9*n)/2) + 1 for all n. - Hugo Pfoertner, Dec 08 2020
From Chai Wah Wu, Aug 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
G.f.: x*(-x^2 + 20*x + 17)/((x - 1)^2*(x + 1)). (End)

A339532 Numbers b > 1 such that the smallest three primes, i.e., 2, 3 and 5 are base-b Wieferich primes.

Original entry on oeis.org

449, 557, 593, 649, 701, 757, 793, 901, 1349, 1457, 1493, 1549, 1601, 1657, 1693, 1801, 2249, 2357, 2393, 2449, 2501, 2557, 2593, 2701, 3149, 3257, 3293, 3349, 3401, 3457, 3493, 3601, 4049, 4157, 4193, 4249, 4301, 4357, 4393, 4501, 4949, 5057, 5093, 5149, 5201
Offset: 1

Views

Author

Felix Fröhlich, Dec 08 2020

Keywords

Crossrefs

Cf. A256236. Row 1 of A319060.
Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Programs

  • Mathematica
    Select[Range[2, 5250], Function[b, AllTrue[{2, 3, 5}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)
  • PARI
    is(n) = forprime(p=1, 5, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

Formula

Conjectures from Chai Wah Wu, Aug 18 2025: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
G.f.: x*(-x^8 + 108*x^7 + 36*x^6 + 56*x^5 + 52*x^4 + 56*x^3 + 36*x^2 + 108*x + 449)/(x^9 - x^8 - x + 1). (End)

A339533 Numbers b > 1 such that the smallest four primes, i.e., 2, 3, 5 and 7 are base-b Wieferich primes.

Original entry on oeis.org

557, 901, 1549, 2449, 4049, 5293, 5849, 6193, 7057, 7957, 8801, 9701, 12349, 13249, 14093, 14993, 15857, 16201, 16757, 18001, 19601, 20501, 21149, 21493, 22049, 23257, 23293, 24893, 25057, 25793, 26657, 27557, 28549, 30349, 31949, 32849, 33301, 34201, 35801
Offset: 1

Views

Author

Felix Fröhlich, Dec 08 2020

Keywords

Crossrefs

Cf. A256236. Row 1 of A319061.
Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339534 (k=5), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Programs

  • Mathematica
    Select[Range[2, 36000], Function[b, AllTrue[{2, 3, 5, 7}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)
  • PARI
    is(n) = forprime(p=1, 7, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339534 Numbers b > 1 such that the smallest five primes, i.e., 2, 3, 5, 7 and 11 are base-b Wieferich primes.

Original entry on oeis.org

19601, 22049, 48149, 52057, 67357, 84457, 85193, 93493, 104057, 113257, 115757, 132857, 160849, 171793, 186101, 198449, 206749, 209249, 224549, 228457, 235493, 261593, 265501, 280801, 297901, 317501, 326701, 329201, 329849, 346301, 359857, 374293, 374849
Offset: 1

Views

Author

Felix Fröhlich, Dec 08 2020

Keywords

Crossrefs

Cf. A256236. Row 1 of A319062.
Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Programs

  • Mathematica
    Select[Range[2, 375000], Function[b, AllTrue[{2, 3, 5, 7, 11}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)
  • PARI
    is(n) = forprime(p=1, 11, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339535 Numbers b > 1 such that the smallest six primes, i.e., 2, 3, 5, 7, 11 and 13 are base-b Wieferich primes.

Original entry on oeis.org

132857, 171793, 261593, 618301, 700993, 997757, 1211201, 1365857, 1388593, 1542293, 1557593, 1681649, 1692557, 1906001, 2086793, 2124757, 2245357, 2293757, 2341349, 2443501, 2822957, 3025457, 3036401, 3435193, 3554657, 3569257, 3814649, 4028093, 4048901
Offset: 1

Views

Author

Felix Fröhlich, Dec 08 2020

Keywords

Crossrefs

Cf. A256236. Row 1 of A319063.
Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339534 (k=5), A339536 (k=7), A339537 (k=8).

Programs

  • Mathematica
    Select[Range[2, 4100000], Function[b, AllTrue[{2, 3, 5, 7, 11, 13}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)
  • PARI
    is(n) = forprime(p=1, 13, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339536 Numbers b > 1 such that the smallest seven primes, i.e., 2, 3, 5, 7, 11, 13 and 17 are base-b Wieferich primes.

Original entry on oeis.org

4486949, 4651993, 4941649, 5571593, 11903257, 19397501, 19841257, 19942001, 20610901, 21530501, 25793893, 42969601, 44404093, 46336949, 51509701, 52786493, 53740457, 54561401, 56120257, 56904857, 63789749, 64028593, 65497301, 65666449, 68441957, 72582101
Offset: 1

Views

Author

Felix Fröhlich, Dec 08 2020

Keywords

Crossrefs

Cf. A256236. Row 1 of A319064.
Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339537 (k=8).

Programs

  • PARI
    is(n) = forprime(p=1, 17, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1
Showing 1-6 of 6 results.

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