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 1-10 of 28 results. Next

A181470 Numbers n such that 97 is the largest prime factor of n^2 - 1.

Original entry on oeis.org

96, 98, 193, 195, 290, 389, 484, 581, 583, 775, 872, 874, 969, 971, 1066, 1163, 1165, 1359, 1456, 1551, 1553, 1648, 1747, 1844, 1939, 2036, 2133, 2135, 2232, 2521, 2715, 2911, 3008, 3103, 3299, 3394, 3396, 3590, 3976, 4267, 4269, 4463, 4558, 4946, 5045
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(25) = 99913980938200001; primepi(97) = 25.

Links

  • Lucas A. Brown, Table of n, a(n) for n = 1..2734 (2679 terms from Artur Jasinski, 55 missing terms from Lucas A. Brown, complete subject to the effective abc conjecture c < rad(abc)^2)
  • From _David A. Corneth_, Oct 03 2022: (Start)
  • To verify full I listed all 97-smooth numbers k that are a multiple of 97 below (inclusive) A175607(25) + 2. I then checked if k+2 is 97-smooth. If so, k+1 is a term. Then similarily I checked if k-2 is 97-smooth. If so, k-1 is a term.
  • Doing so found the 2734 terms from the b-file. All candidates have been checked completing the proof of full. (End)

Crossrefs

Cf. A175607, A181447-A181469, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 97 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 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 97 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 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 == 97, AppendTo[rr, n]]]; n++]; rr
(* or *)
Select[Range[300000], FactorInteger[#^2 - 1][[-1, 1]] == 97 &]
  • PARI
    is(n)=n=n^2-1;forprime(p=2,89,n/=p^valuation(n,p));n>1 && 97^valuation(n,97)==n \\ Charles R Greathouse IV, Jul 01 2013

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*)

A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

16, 33, 35, 50, 67, 69, 101, 103, 118, 120, 169, 188, 239, 271, 307, 339, 441, 511, 545, 577, 749, 883, 1121, 1189, 1376, 1429, 1665, 1871, 2024, 2177, 2311, 2449, 2549, 3401, 4115, 4861, 4999, 5201, 9827, 11663, 24751, 28799, 57121, 62425, 74359, 388961, 672281
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 17.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(7) = 672281; primepi(17) = 7.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181451, A181453-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..350000] | m eq 17 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..700000] | p mod (n^2-1) eq 0 and (D[#D] eq 17 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 < 700000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 17, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[680000], FactorInteger[#^2-1][[-1, 1]]==17&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 13, n/=p^valuation(n, p)); n>1 && 17^valuation(n, 17)==n \\ Charles R Greathouse IV, Jul 01 2013

A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

18, 20, 37, 39, 56, 77, 113, 134, 151, 153, 170, 191, 246, 265, 305, 324, 341, 362, 379, 417, 419, 571, 626, 647, 664, 685, 721, 799, 911, 951, 989, 1025, 1616, 1937, 2431, 2661, 2889, 3041, 3079, 3212, 3457, 3970, 4751, 4863, 5851, 6271, 6499, 8399, 11551, 11857
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 19.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(8) = 23718421; primepi(19) = 8.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181452, A181454-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 19 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..24000000] | p mod (n^2-1) eq 0 and (D[#D] eq 19 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 < 24000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 19, AppendTo[rr, n]]]; n++ ]; rr
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==19&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 17, n/=p^valuation(n, p)); n>1 && 19^valuation(n, 19)==n \\ Charles R Greathouse IV, Jul 01 2013

A181455 Numbers k such that 29 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

28, 57, 59, 86, 115, 144, 146, 175, 231, 233, 289, 349, 376, 407, 436, 463, 494, 521, 579, 639, 666, 755, 811, 987, 1101, 1103, 1217, 1275, 1451, 1565, 1567, 1681, 2029, 2089, 2551, 2872, 2899, 3191, 3249, 3365, 4001, 4003, 4351, 4409, 4523, 4929, 5279
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 29.
Sequence is finite. For proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(10) = 354365441; primepi(29) = 10.

Links

Crossrefs

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

Programs

  • Magma
    [ n: n in [2..6000] | m eq 29 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011
  • Mathematica
    Select[Range[5600],FactorInteger[#^2-1][[-1,1]]==29&]  (* Harvey P. Dale, Feb 16 2011 *)
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 23, n/=p^valuation(n, p)); n>1 && 29^valuation(n, 29)==n \\ Charles R Greathouse IV, Jul 01 2013

A181457 Numbers k such that 37 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

36, 38, 73, 75, 149, 186, 221, 223, 260, 295, 369, 371, 406, 443, 482, 519, 593, 628, 776, 813, 815, 961, 1000, 1072, 1259, 1331, 1333, 1405, 1407, 1444, 1481, 1701, 1814, 1849, 1886, 1923, 1999, 2071, 2367, 2591, 2663, 2737, 2887, 2959, 3329, 3331, 3403
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 37.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(12) = 9447152318; primepi(37) = 12.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181456, A181458-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 37 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 37 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 == 37, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==37&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 31, n/=p^valuation(n, p)); n>1 && 37^valuation(n, 37)==n \\ Charles R Greathouse IV, Jul 01 2013

A181458 Numbers k such that 41 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

40, 81, 83, 122, 124, 163, 204, 206, 247, 286, 288, 329, 409, 491, 493, 573, 575, 737, 739, 778, 901, 944, 985, 1024, 1065, 1106, 1149, 1231, 1393, 1518, 1559, 1639, 1682, 2049, 2051, 2092, 2295, 2377, 2379, 2623, 2705, 2789, 3035, 3158, 3199, 3361, 3363
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 41.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(13) = 127855050751; primepi(41) = 13.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181457, A181459-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 41 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 41 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 == 41, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==41&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 37, n/=p^valuation(n, p)); n>1 && 41^valuation(n, 41)==n \\ Charles R Greathouse IV, Jul 01 2013

A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

42, 44, 85, 87, 171, 173, 216, 257, 259, 300, 343, 386, 431, 474, 517, 560, 601, 687, 689, 730, 818, 859, 1074, 1117, 1119, 1289, 1291, 1332, 1420, 1549, 1633, 1721, 1805, 1891, 1977, 1979, 2108, 2321, 2495, 2665, 2667, 2751, 2753, 2794, 2925, 3095, 3484
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 43.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(14) = 842277599279; primepi(43) = 14.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181458, A181460-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 43 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 43 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 == 43, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==43&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 41, n/=p^valuation(n, p)); n>1 && 43^valuation(n, 43)==n \\ Charles R Greathouse IV, Jul 01 2013
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