A181470 -id:A181470 - cp's OEIS frontend

A181447 Numbers k such that 3 is the largest prime factor of k^2 - 1.

2, 5, 7, 17

Offset: 1

Artur Jasinski, Oct 21 2010

Sequence is finite and complete, for proof see A175607.

Search for terms can be restricted to the range from 2 to A175607(2) = 17; primepi(3) = 2.

[ n: n in [2..20] | m eq 3 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011

Select[Range[20], FactorInteger[#^2-1][[-1, 1]]==3&]

is(n)=n=n^2-1;my(o=valuation(n,3)); o && n>>valuation(n/3^o,2)==1 \\ 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

Klaus Brockhaus, Oct 31 2010

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)

David A. Corneth, Table of n, a(n) for n = 1..3340 (first 2325 terms from Klaus Brockhaus, terms < 10^17).
Florian Luca, Filip Najman, On the largest prime factor of x^2-1, arXiv:1005.1533 [math.NT], 2010.
Florian Luca, Filip Najman, On the largest prime factor of x^2-1, Mathematics of Computation 80 (2011), 429-435. (Paper has errata that was posted on the MOC website.)

Cf. A175607, A181471, A181447-A181470.

[ n: n in [2..5000] | m eq 101 where m is D[#D] where D is PrimeDivisors(n^2-1) ];

Select[Range[4746], FactorInteger[#^2-1][[-1, 1]]==101&] (* Metin Sariyar, Sep 15 2019 *)

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

T. D. Noe, Apr 03 2013

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.

			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}

Andrew Howroyd, Table of n, a(n) for n = 1..16223 (first 25 rows for primes up to 97)
Florian Luca and Filip Najman, On the largest prime factor of x^2-1, arXiv:1005.1533 [math.NT], 2010.
Florian Luca and Filip Najman, On the largest prime factor of x^2-1, Mathematics of Computation 80 (2011), 429-435. (Paper has errata that was posted on the MOC website.)
Filip Najman, List of Publications Page (Adjacent to entry number 4 are links with the data files for the first 25 rows (=16223 terms) of this sequence)

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).

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

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

Artur Jasinski, Oct 21 2010

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.

Artur Jasinski, Table of n, a(n) for n = 1..47

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

[ 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

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

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&]

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

Artur Jasinski, Oct 21 2010

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.

Artur Jasinski, Table of n, a(n) for n = 1..72

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

[ 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

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

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&]

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

Artur Jasinski, Oct 21 2010

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.

Artur Jasinski, Table of n, a(n) for n = 1..126

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

[ 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

Select[Range[5600],FactorInteger[#^2-1][[-1,1]]==29&]  (* Harvey P. Dale, Feb 16 2011 *)

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

Artur Jasinski, Oct 21 2010

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.

Artur Jasinski, Table of n, a(n) for n = 1..208

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

[ 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

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

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&]

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

Artur Jasinski, Oct 21 2010

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.

Artur Jasinski, Table of n, a(n) for n = 1..262

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

[ 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

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

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&]

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

Artur Jasinski, Oct 21 2010

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.

Artur Jasinski, Table of n, a(n) for n = 1..343

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

[ 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

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

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&]

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

A181460 Numbers k such that 47 is the largest prime factor of k^2-1.

Original entry on oeis.org

46, 48, 93, 95, 142, 187, 189, 281, 375, 377, 424, 469, 610, 657, 659, 704, 751, 753, 892, 988, 1033, 1035, 1082, 1174, 1223, 1270, 1364, 1409, 1597, 1599, 1691, 1693, 1926, 1973, 1975, 2022, 2069, 2161, 2255, 2351, 2443, 2584, 2727, 2913, 2915, 3009

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 47.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(15) = 2218993446251; primepi(47) = 15.

Artur Jasinski, Table of n, a(n) for n = 1..433

Cf. A076605, A175607, A181447-A181459, A181461-A181470, A181568.

[ n: n in [2..300000] | m eq 47 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==47&]

is(n)=n=n^2-1; forprime(p=2, 43, n/=p^valuation(n, p)); n>1 && 47^valuation(n, 47)==n \\ Charles R Greathouse IV, Jul 01 2013

cp's OEIS Frontend

A181447 Numbers k such that 3 is the largest prime factor of k^2 - 1.

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

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

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

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

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

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

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