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.

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A181451 Numbers k such that 13 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

12, 14, 25, 27, 51, 53, 64, 79, 129, 131, 155, 181, 209, 274, 287, 337, 391, 649, 701, 703, 727, 846, 1249, 1351, 1457, 1574, 2001, 3431, 4159, 8191, 8449, 13311, 21295, 246401
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 13.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(6) = 246401; primepi(13) = 6.

Crossrefs

Row 6 of A223701.
Cf. A076605, A175607, A181447-A181450, A181452-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..250000] | m eq 13 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 18 2011
  • Mathematica
    Select[Range[250000], FactorInteger[#^2-1][[-1, 1]]==13&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 11, n/=p^valuation(n, p)); n>1 && 13^valuation(n, 13)==n \\ Charles R Greathouse IV, Jul 01 2013

A181454 Numbers k such that 23 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

22, 24, 45, 47, 91, 116, 137, 139, 183, 208, 229, 254, 298, 321, 323, 344, 415, 461, 505, 551, 599, 645, 781, 783, 919, 967, 1013, 1057, 1126, 1151, 1310, 1471, 1519, 1749, 1793, 2186, 2209, 2276, 2393, 2575, 2874, 2991, 3704, 3725, 4047, 4049, 4369
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 23.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(9) = 10285001; primepi(23) = 9.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181453, A181455-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 23 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 18 2011
  • Magma
    p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..10300000] | p mod (n^2-1) eq 0 and (D[#D] eq 23 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 24 2011
  • Mathematica
    jj=2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr ={};n = 2; While[n < 14000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 23, AppendTo[rr, n]]]; n++ ]; rr
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==23&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 19, n/=p^valuation(n, p)); n>1 && 23^valuation(n, 23)==n \\ Charles R Greathouse IV, Jul 01 2013

A181456 Numbers k such that 31 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

30, 32, 61, 63, 92, 94, 125, 154, 185, 249, 309, 311, 342, 373, 404, 433, 495, 526, 528, 559, 681, 683, 714, 869, 898, 929, 991, 1055, 1084, 1177, 1241, 1301, 1427, 1520, 1611, 1673, 1735, 1799, 1861, 1921, 1954, 2047, 2107, 2419, 2696, 2729, 2851, 3037
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 31.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(11) = 3222617399; primepi(31) = 11.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181455, A181457-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 31 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011
  • Magma
    p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 31 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 31, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[5000],Max[Transpose[FactorInteger[ #^2-1]][[1]]]==31&] (* Harvey P. Dale, Nov 03 2010 *)
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 29, n/=p^valuation(n, p)); n>1 && 31^valuation(n, 31)==n \\ Charles R Greathouse IV, Jul 01 2013

A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

Original entry on oeis.org

2, 10, 108, 235, 1201, 390050, 314766, 4035, 364384, 50411, 25955045, 5254864, 236558593, 16958526, 20388056, 177544434, 492981885, 2275400230, 256347346, 384902923486, 324850200677887
Offset: 1

Views

Author

Charles R Greathouse IV, Feb 21 2011

Keywords

Comments

For any prime p, there are finitely many x such that x^2-2 has p as its largest prime factor.

Links

Crossrefs

Cf. A242488, A379348.
Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1).

Extensions

a(21) added by Andrew Howroyd, Dec 22 2024

A379344 a(n) is the sum of all numbers k such that the greatest prime factor of k^2 - 1 is prime(n).

Original entry on oeis.org

3, 31, 310, 15055, 30433, 318914, 1378856, 41139929, 29628346, 706390476, 5330866189, 17573061167, 227644494516, 912323845104, 3312744735567, 6366920047986, 69033389180772, 89835379146224, 45938747179900, 564448183072697, 6856082910702485, 19187647510345764511, 56226662050090628, 357824287346707561, 139924756071743686
Offset: 1

Views

Author

Andrew Howroyd, Dec 21 2024

Keywords

Comments

See A175607 and A223701 for additional information.

Crossrefs

Row sums of A223701.
Cf. A175607.

Formula

a(n) = Sum_{i>=1} A223701(n,i).

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

A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

22, 140, 707, 21362, 4991, 1306066, 137965, 2294636, 31768298, 1557652, 340064590, 38439662, 105080665, 273502688, 543164542, 9575480365630, 391890109484, 14629598023, 80849485336, 1241646894380
Offset: 1

Views

Author

Charles R Greathouse IV, Feb 21 2011

Keywords

Comments

For any prime p, there are finitely many x such that x^2+2 has p as its largest prime factor.

Links

Crossrefs

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1), A185396 (x^2 - 2).

A223703 Conjectured irregular triangle (with some rows blank) of numbers k such that prime(n) is the largest prime factor of k^3 - 1.

Original entry on oeis.org

2, 4, 3, 9, 16, 22, 18, 7, 11, 30, 5, 25, 67, 191, 10, 26, 100, 121, 211, 581, 676, 6, 36, 49, 79, 87, 165, 6205, 178, 13, 47, 501, 562, 29, 37, 68, 135, 163, 565, 900, 1369, 1712, 3446, 4624, 8, 64, 74, 81, 137, 373, 439, 1451, 1816, 2629, 7527, 39209
Offset: 1

Views

Author

T. D. Noe, Apr 03 2013

Keywords

Comments

Primes 2, 3, 5, 11, 23, 41, 53, 71, 83, 89,... do not appear as largest factors. However, they can be smaller factors. For instance, 3^3 - 1 = 2 * 13.

Examples

			Irregular triangle:
2:  {},
3:  {},
5:  {},
7:  {2, 4},
11: {},
13: {3, 9, 16, 22},
17: {18},
19: {7, 11},
23: {},
29: {30},
31: {5, 25, 67, 191},
37: {10, 26, 100, 121, 211, 581, 676},
41: {},
43: {6, 36, 49, 79, 87, 165},
47: {6205},
53: {},
59: {178},
61: {13, 47, 501, 562},
67: {29, 37, 68, 135, 163, 565, 900, 1369, 1712, 3446, 4624},
71: {},
73: {8, 64, 74, 81, 137, 373, 439, 1451, 1816, 2629, 7527, 39209}

Crossrefs

Cf. A175607 (largest number k such that the greatest prime factor of k^2-1 is prime(n)).
Cf. A223701-A223707 (related sequences).

Programs

  • Mathematica
    t = Table[FactorInteger[n^3 - 1][[-1,1]], {n, 2, 10^6}]; Table[1 + Flatten[Position[t, Prime[n]]], {n, 25}]

A223704 Conjectured irregular triangle (with some rows blank) of numbers k such that prime(n) is the largest prime factor of k^3 + 1.

Original entry on oeis.org

1, 2, 3, 5, 19, 4, 10, 17, 23, 8, 12, 31, 69, 6, 26, 68, 11, 27, 101, 122, 7, 37, 50, 80, 179, 582, 14, 48, 75, 563, 719, 2820, 4135, 30, 38, 164, 231, 440, 566, 901, 11093, 112925, 267167, 212, 9, 65, 374, 20303, 24, 56, 103, 293, 530, 656, 767, 868, 82, 2157
Offset: 1

Views

Author

T. D. Noe, Apr 03 2013

Keywords

Comments

Primes 5, 11, 17, 23, 29, 47, 59,... do not appear as largest factors. However, they can be smaller factors. For instance, 4^3 + 1 = 5 * 13.

Examples

			Irregular triangle:
2:  {1},
3:  {2},
5:  {},
7:  {3, 5, 19},
11: {},
13: {4, 10, 17, 23},
17: {},
19: {8, 12, 31, 69},
23: {},
29: {},
31: {6, 26, 68},
37: {11, 27, 101},
41: {122},
43: {7, 37, 50, 80, 179},
47: {},
53: {582},
59: {},
61: {14, 48, 75, 563, 719, 2820, 4135},
67: {30, 38, 164, 231, 440, 566, 901, 11093, 112925, 267167},
71: {212},
73: {9, 65, 374, 20303},
79: {24, 56, 103, 293, 530, 656, 767, 868},
83: {82, 2157}.

Crossrefs

Cf. A175607 (largest number k such that the greatest prime factor of k^2-1 is prime(n)).
Cf. A223701-A223707 (related sequences).

Programs

  • Mathematica
    t = Table[FactorInteger[n^3 + 1][[-1,1]], {n, 10^6}]; Table[Flatten[Position[t, Prime[n]]], {n, 25}]

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
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