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.

A181568 Numbers k such that the largest prime factor of k^2-1 is 101.

Original entry on oeis.org

100, 201, 203, 302, 304, 403, 405, 506, 607, 706, 807, 809, 1009, 1011, 1112, 1211, 1312, 1415, 1514, 1516, 1716, 1819, 1918, 2221, 2324, 2524, 2526, 2625, 2627, 3231, 3233, 3334, 3433, 3635, 3736, 3839, 4041, 4241, 4344, 4445, 4544, 4645, 4647, 4746
Offset: 1

Views

Author

Klaus Brockhaus, Oct 31 2010

Keywords

Comments

Sequence is finite, number of terms and last term are still unknown (cf. A175607, A181471).
From David A. Corneth, Sep 11 2019: (Start)
Are there any terms > 941747621709311?
As k^2 - 1 = (k - 1)(k + 1), a(n) is of the form 101*m +- 1. (End)

Links

Crossrefs

Cf. A175607, A181471, A181447-A181470.

Programs

  • Magma
    [ n: n in [2..5000] | m eq 101 where m is D[#D] where D is PrimeDivisors(n^2-1) ];
  • Mathematica
    Select[Range[4746], FactorInteger[#^2-1][[-1, 1]]==101&] (* Metin Sariyar, Sep 15 2019 *)
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 97, n/=p^valuation(n, p)); n>1 && 101^valuation(n, 101)==n \\ Charles R Greathouse IV, Jul 01 2013

A223701 Irregular triangle of numbers k such that prime(n) is the largest prime factor of k^2 - 1.

Original entry on oeis.org

3, 2, 5, 7, 17, 4, 9, 11, 19, 26, 31, 49, 161, 6, 8, 13, 15, 29, 41, 55, 71, 97, 99, 127, 244, 251, 449, 4801, 8749, 10, 21, 23, 34, 43, 65, 76, 89, 109, 111, 197, 199, 241, 351, 485, 769, 881, 1079, 6049, 19601, 12, 14, 25, 27, 51, 53, 64, 79, 129, 131, 155
Offset: 1

Views

Author

T. D. Noe, Apr 03 2013

Keywords

Comments

Note that the first number of each row forms the sequence 3, 2, 4, 6, 10, 12,..., which is A039915. The first 25 rows, except the first, are in A181447-A181470.

Examples

			Irregular triangle:
  {3},
  {2, 5, 7, 17},
  {4, 9, 11, 19, 26, 31, 49, 161},
  {6, 8, 13, 15, 29, 41, 55, 71, 97, 99, 127, 244, 251, 449, 4801, 8749}

Links

Crossrefs

Rows 2..25 with complete data are: A181447, A181448, A181449, A181450, A181451, A181452, A181453, A181454, A181455, A181456, A181457, A181458, A181459, A181460, A181461, A181462, A181463, A181464, A181465, A181466, A181467, A181468, A181469, A181470.
Row 26 is A181568.
Cf. A039915 (first terms), A175607 (last terms), A181471 (row lengths), A379344 (row sums).
Cf. A223702, A223703, A223704 (related tables).

Programs

  • Mathematica
    t = Table[FactorInteger[n^2 - 1][[-1,1]], {n, 2, 10^5}]; Table[1 + Flatten[Position[t, Prime[n]]], {n, 6}]

A175904 Numbers m for which the set of prime divisors of m^2-1 is unique.

Original entry on oeis.org

2, 3, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 36, 38, 39, 40, 42, 44, 45, 46, 47, 48, 50, 52, 54, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 73, 74, 75, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 93, 94, 95, 96, 98, 99
Offset: 1

Views

Author

Artur Jasinski, Oct 12 2010, Oct 21 2010

Keywords

Comments

Complement of A175903. A proof for the presence of the first 63 terms (for which the largest prime divisor is < 100) follows along the lines of the comment in A175607.

Examples

			The unique prime factor sets are {3} (m=2), {2} (m=3), {5,7} (m=6), {3,7} (m=8), {2,5} (m=9) etc.

Crossrefs

Cf. A175607, A175901 - A175904, A181447 - A181471.

Programs

  • Mathematica
    aa = {}; bb = {}; cc = {}; ff = {}; Do[k = n^2 - 1; kk = FactorInteger[k]; b = {}; Do[AppendTo[b, kk[[m]][[1]]], {m, 1, Length[kk]}]; dd = Position[aa, b]; If[dd == {}, AppendTo[cc, n]; AppendTo[aa, b], AppendTo[ff, n]; AppendTo[bb, cc[[dd[[1]][[1]]]]]], {n, 2, 1000000}]; jj=Table[n,{2,99}]; ss=Union[bb,ff]; Take[Complement[jj,ss],63] (*Artur Jasinski*)

A175902 Values of k in A175901.

Original entry on oeis.org

5, 5, 11, 4, 11, 29, 11, 25, 13, 23, 29, 34, 13, 89, 13, 51, 11, 151, 43, 89, 181, 169, 89, 29, 101, 59, 223, 111, 181, 269, 125, 29, 23, 101, 83, 35, 56, 305, 79, 113, 181, 287, 151, 155, 379, 349, 769, 545, 329, 505, 571, 37, 373, 769, 344, 91, 1121, 79, 353, 79, 985
Offset: 1

Views

Author

Artur Jasinski, Oct 11 2010, Oct 21 2010

Keywords

Crossrefs

Cf. A175607, A175901-A175904, A181447-A181471.

Programs

  • PARI
    isok(n) = {pfs = factor(n^2-1)[,1]; for (k = 2, n-1, if (factor(k^2-1)[,1] == pfs, return (k));); return (0);}
lista(nn) = {for(n=2, nn, if (k = isok(n), print1(k, ", ");););} \\ Michel Marcus, Nov 04 2013

Extensions

Edited by N. J. A. Sloane, Oct 14 2010

A379349 Number of integers of the form k^2 + 2 whose greatest prime factor is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

1, 5, 5, 7, 11, 12, 18, 18, 21, 25, 30, 47, 39, 45, 62, 63, 83, 81, 107, 105, 130
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2024

Keywords

Comments

See A379350 for additional information.

Examples

			Table showing n, p = A033203(n) and a(n):
   1    2    1
   2    3    5
   3   11    5
   4   17    7
   5   19   11
   6   41   12
   7   43   18
   8   59   18
   9   67   21
  10   73   25
  ...

Links

Crossrefs

Row lengths of A379350.
Cf. A033203, A181471, A379346, A379348.

A175903 Numbers n such that there is another number k such that n^2-1 and k^2-1 have the same set of prime factors.

Original entry on oeis.org

4, 5, 7, 11, 13, 17, 19, 23, 25, 26, 29, 31, 34, 35, 37, 41, 43, 49, 51, 53, 55, 56, 59, 61, 65, 67, 71, 76, 79, 81, 83, 89, 91, 92, 97, 101, 109, 111, 113, 125, 127, 129, 131, 139, 149, 151, 155, 161, 169, 179, 181, 187, 191, 197, 199, 209, 223, 235, 239, 241, 251
Offset: 1

Views

Author

Artur Jasinski, Oct 12 2010, Oct 21 2010

Keywords

Comments

The difference from A175901 is that k may also be larger than n. So we obtain the sequence by building the union of the sets A175901 and A175902, and sorting.

Examples

			a(2)=5 because set of prime divisors of 5^2-1 =2^3*3 is {2,3}, the same as for example for 7^2-1 = 2^4*3.

Crossrefs

Cf. A175607, A175901-A175904, A181447-A181471.

Programs

  • Mathematica
    aa = {}; bb = {}; cc = {}; ff = {}; Do[k = n^2 - 1; kk = FactorInteger[k]; b = {}; Do[AppendTo[b, kk[[m]][[1]]], {m, 1, Length[kk]}]; dd = Position[aa, b]; If[dd == {}, AppendTo[cc, n]; AppendTo[aa, b], AppendTo[ff, n]; AppendTo[bb, cc[[dd[[1]][[1]]]]]], {n, 2, 1000000}]; Take[Union[bb,ff],100] (* Artur Jasinski *)

Extensions

Name improved by T. D. Noe, Nov 15 2010

A379345 Number of integers of the form k^2 - 1 whose greatest prime factor is at most prime(n).

Original entry on oeis.org

1, 5, 13, 29, 49, 83, 130, 202, 297, 423, 591, 799, 1061, 1404, 1837, 2344, 2978, 3777, 4753, 5899, 7338, 9036, 11118, 13489, 16223
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2024

Keywords

Comments

See A181471 and A223701 for additional information.

Links

Crossrefs

Partial sums of A181471.
Cf. A175607, A223701, A379344.
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.