A223701 -id:A223701 - cp's OEIS frontend

A242488 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 - 2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

2, 3, 4, 10, 6, 11, 45, 108, 5, 18, 28, 74, 156, 235, 8, 23, 39, 116, 1201, 17, 24, 58, 147, 304, 550, 2272, 390050, 7, 40, 54, 87, 101, 181, 557, 1558, 43764, 314766, 12, 59, 130, 225, 414, 1077, 1124, 2686, 3420, 4035, 32, 41, 178, 333, 698, 844, 1638, 4567, 15362, 364384

Offset: 1

Juri-Stepan Gerasimov, May 16 2014

From Andrew Howroyd, Dec 22 2024: (Start)
For any prime p, there are finitely many x such that x^2 - 2 has p as its largest prime factor.
The Filip Najman data file gives all 537 numbers x such that x^2 - 2 has no prime factor greater than 199. This includes a value for x = 1 which is not included here. (End)

			Triangle of numbers k such that p is the greatest prime factor of k^2 - 2:
p\k  |  1 |  2 |  3  |  4  |  5   |  6   |  7   |  >= 8
------------------------------------------------------------------------
   2 |  2 |    |     |     |      |      |      |
   7 |  3 |  4 |  10 |     |      |      |      |
  17 |  6 | 11 |  45 | 108 |      |      |      |
  23 |  5 | 18 |  28 |  74 |  156 |  235 |      |
  31 |  8 | 23 |  39 | 116 | 1201 |      |      |
  41 | 17 | 24 |  58 | 147 |  304 |  550 | 2272 | 390050;
  47 |  7 | 40 |  54 |  87 |  101 |  181 |  557 | 1558, 43764, 314766;
  71 | 12 | 59 | 130 | 225 |  414 | 1077 | 1124 | 2686, 3420, 4035;
  73 | 32 | 41 | 178 | 333 |  698 |  844 | 1638 | 4567, 15362, 364384;
  ...
6 is a term of row 3 because (6^2 - 2)/17 = 2 and 2 < 17;
11 is a term of row 3 because (11^2 - 2)/17 = 7 and 7 < 17;
45 is a term of row 3 because (45^2 - 2)/17^2 = 7 and 7 < 17;
108 is a term of row 3 because (108^2 - 2)/17 = 686 = 2*7^3 and 7 < 17.

Andrew Howroyd, Table of n, a(n) for n = 1..536 (first 21 rows for primes up to 199)
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for the first 21 rows of this sequence).

Cf. A038873, A164314, A059770 (first terms for n>1), A185396 (last terms), A379348 (row lengths).

Cf. A223701.

Converted to triangle by Andrew Howroyd, Dec 22 2024

A181471 a(n) = number of numbers of the form k^2-1 having n-th prime as largest prime divisor.

Original entry on oeis.org

1, 4, 8, 16, 20, 34, 47, 72, 95, 126, 168, 208, 262, 343, 433, 507, 634, 799, 976, 1146, 1439, 1698, 2082, 2371, 2734

Offset: 1

Artur Jasinski, Oct 21-22 2010

Theorem: zero does not occur in this sequence. Proof: (p-1)^2-1=(p-2)p. This means that p is greatest prime divisor of (p-1)^2-1 for every p.

An effective abc conjecture (c < rad(abc)^2) would imply that a(24)-a(33) are (2371, 2734, 3360, 4022, 4637, 5575, 6424, 7268, 8351, 9661). - Lucas A. Brown, Oct 01 2022

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.)
Wikipedia, Størmer's theorem.

Row lengths of A223701.

Cf. A039915, A175607.

Wrong terms a(24)-a(25) removed by Lucas A. Brown, Oct 01 2022

a(24)-a(25) from David A. Corneth, Oct 01 2022

A223702 Irregular triangle of numbers k such that A002313(n), the n-th prime not congruent to 3 mod 4 is the largest prime factor of k^2 + 1.

Original entry on oeis.org

1, 2, 3, 7, 5, 8, 18, 57, 239, 4, 13, 21, 38, 47, 268, 12, 17, 41, 70, 99, 157, 307, 6, 31, 43, 68, 117, 191, 302, 327, 882, 18543, 9, 32, 73, 132, 278, 378, 829, 993, 2943, 23, 30, 83, 182, 242, 401, 447, 606, 931, 1143, 1772, 6118, 34208, 44179, 85353, 485298

Offset: 1

T. D. Noe, Apr 03 2013

Note that primes of the form 4x+3 are not divisors.

			Irregular triangle:
   p | {k}
-----+---------------------------------
   2 | {1},
   5 | {2, 3, 7},
  13 | {5, 8, 18, 57, 239},
  17 | {4, 13, 21, 38, 47, 268},
  29 | {12, 17, 41, 70, 99, 157, 307},
  37 | {6, 31, 43, 68, 117, 191, 302, 327, 882, 18543},
  41 | {9, 32, 73, 132, 278, 378, 829, 993, 2943}
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..811 (first 22 rows for primes up to 197)
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for the first 22 rows (=811 terms) of this sequence). [As of Dec 2024, the link has the incorrect URL. Should be https://web.math.pmf.unizg.hr/~fnajman/rezplus1.html]

Cf. A002313, A014442, A177979 (first terms), A185389 (last terms), A223705, A285283, A379346 (row lengths), A379347 (row sums).
Cf. A285282, A285523.
Cf. A223701, A223703, A223704 (related tables).

t = Table[FactorInteger[n^2 + 1][[-1,1]], {n, 10^5}]; Table[Flatten[Position[t, Prime[n]]], {n, 13}]

Definition amended by Andrew Howroyd, Dec 22 2024

A181465 Numbers k such that 71 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

70, 141, 143, 214, 283, 285, 356, 425, 496, 569, 638, 709, 780, 782, 851, 853, 924, 993, 1135, 1208, 1277, 1279, 1561, 1563, 1703, 1847, 2058, 2129, 2131, 2344, 2413, 2626, 2699, 2839, 2841, 3054, 3265, 3267, 3336, 3338, 3409, 3478, 3480, 3551, 3620, 3691

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 71.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(20) = 238178082107393; primepi(71) = 20.

Artur Jasinski, Table of n, a(n) for n = 1..1146

Row 20 of A223701.

Cf. A076605, A175607, A181447-A181464, A181466-A181470, A181568.

[ n: n in [2..300000] | m eq 71 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 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 < 14000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 71, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==71&]

is(n)=n=n^2-1; forprime(p=2, 67, n/=p^valuation(n, p)); n>1 && 71^valuation(n, 71)==n \\ Charles R Greathouse IV, Jul 01 2013

A181466 Numbers k such that 73 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

72, 74, 145, 147, 218, 220, 291, 293, 364, 439, 512, 729, 731, 804, 875, 1021, 1023, 1167, 1169, 1240, 1313, 1315, 1459, 1461, 1607, 1678, 1680, 1751, 1826, 1899, 2045, 2116, 2262, 2481, 2483, 2554, 2702, 2773, 2848, 3067, 3284, 3359, 3576, 3649, 3722

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 73.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(21) = 4573663454608289; primepi(73) = 21.

Artur Jasinski, Table of n, a(n) for n = 1..1439

Row 21 of A223701

Cf. A076605, A175607, A181447-A181465, A181467-A181470, A181568.

[ n: n in [2..300000] | m eq 73 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 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 73 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 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 == 73, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==73&]

is(n)=n=n^2-1; forprime(p=2, 71, n/=p^valuation(n, p)); n>1 && 73^valuation(n, 73)==n \\ Charles R Greathouse IV, Jul 01 2013

A181467 Numbers k such that 79 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

78, 80, 157, 159, 236, 317, 473, 475, 552, 554, 631, 712, 791, 868, 870, 947, 949, 1026, 1028, 1105, 1184, 1421, 1737, 1739, 1816, 1897, 2053, 2134, 2211, 2213, 2369, 2450, 2529, 2685, 2687, 2843, 2924, 3001, 3161, 3477, 3554, 3870, 3949, 3951, 4186, 4188

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 79.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(22) = 19182937474703818751; primepi(79) = 22.

Artur Jasinski, Table of n, a(n) for n = 1..1698

Row 22 of A223701.

Cf. A076605, A175607, A181447-A181466, A181468-A181470, A181568.

[ n: n in [2..300000] | m eq 79 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 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 79 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 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 == 79, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==79&]

is(n)=n=n^2-1; forprime(p=2, 73, n/=p^valuation(n, p)); n>1 && 79^valuation(n, 79)==n \\ Charles R Greathouse IV, Jul 01 2013

A181468 Numbers k such that 83 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

82, 84, 165, 167, 248, 331, 414, 497, 499, 582, 665, 829, 831, 914, 995, 1080, 1161, 1246, 1327, 1329, 1495, 1576, 1825, 1910, 2076, 2157, 2159, 2323, 2406, 2408, 2738, 2821, 2906, 2989, 3070, 3238, 3319, 3485, 3568, 3651, 3653, 3817, 4149, 4234, 4481

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 83.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(23) = 34903240221563713 = a(2082); pi(83) = 23.

Artur Jasinski, Table of n, a(n) for n = 1..2082

Row 23 of A223701.

Cf. A000720, A076605, A175607, A181447-A181467, A181469-A181470, A181568.

[ n: n in [2..300000] | m eq 83 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 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 83 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 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 == 83, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==83&]

is(n)=n=n^2-1; forprime(p=2, 79, n/=p^valuation(n, p)); n>1 && 83^valuation(n, 83)==n \\ Charles R Greathouse IV, Jul 01 2013

A223707 Conjectured least number k such that prime(n) is the largest divisor of k^3 + 1, or 0 if there is no such k.

Original entry on oeis.org

1, 2, 0, 3, 0, 4, 0, 8, 0, 0, 6, 11, 122, 7, 0, 582, 0, 14, 30, 212, 9, 24, 82, 88, 36, 1817, 47, 0, 46, 677, 20, 654, 136, 43, 2383, 33, 13, 59, 166, 1037, 210682, 49, 381, 85, 23245, 93, 15, 40, 18613, 95, 5591, 1433, 16, 0, 1798, 788, 26361, 29, 117, 842

Offset: 1

T. D. Noe, Apr 03 2013

nonn

We allowed k to vary up to 10^7.

Cf. A223701-A223706 (related sequences).

nn = 60; t = Table[0, {nn}]; ps = Prime[Range[nn]]; Do[num = n^3 + 1; j = 0; lastP = 0; While[num > 0 && j < nn, j++; p = ps[[j]]; While[Mod[num, p] == 0, lastP = j; num = num/p]];If[num == 1 && t[[lastP]] == 0, t[[lastP]] = n; Print[{lastP, n}]], {n, 10^7}]; t

A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

0, 1, 2, 4, 5, 22, 3, 8, 14, 19, 140, 7, 10, 24, 41, 58, 265, 707, 6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362, 11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991, 16, 27, 59, 70, 102, 113, 317, 500, 586, 1048, 2951, 3424, 4972, 8240, 12658, 83834, 686210, 1306066

Offset: 1

Andrew Howroyd, Dec 22 2024

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

			Irregular triangle begins:
   p | {k}
-----+------------------
   2 | {0}
   3 | {1, 2, 4, 5, 22}
  11 | {3, 8, 14, 19, 140}
  17 | {7, 10, 24, 41, 58, 265, 707}
  19 | {6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362}
  41 | {11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991}
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..915 (first 21 rows for primes up to 193)
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for rows 2..21 (=914 terms) of this sequence).

Cf. A033203, A379351, A379352 (first terms), A185397 (last terms), A379349 (row lengths).

Cf. A223701, A223702, A242488.

A181448 Numbers k such that 5 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

4, 9, 11, 19, 26, 31, 49, 161

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 3.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(3) = 161; primepi(5) = 3.

Row 3 of A223701.

Cf. A076605, A175607, A181447, A181449-A181470, A181568.

[ n: n in [2..200] | m eq 5 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011

Select[Range[200], FactorInteger[#^2-1][[-1, 1]]==5&]

is(n)=n=n^2-1; n>>=valuation(n,2); n/=3^valuation(n,3); n>1 && 5^valuation(n, 5)==n \\ Charles R Greathouse IV, Jul 01 2013

cp's OEIS Frontend

A242488 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 - 2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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A181471 a(n) = number of numbers of the form k^2-1 having n-th prime as largest prime divisor.

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A223702 Irregular triangle of numbers k such that A002313(n), the n-th prime not congruent to 3 mod 4 is the largest prime factor of k^2 + 1.

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A181465 Numbers k such that 71 is the largest prime factor of k^2 - 1.

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A181466 Numbers k such that 73 is the largest prime factor of k^2 - 1.

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Magma

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A181467 Numbers k such that 79 is the largest prime factor of k^2 - 1.

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Magma

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A181468 Numbers k such that 83 is the largest prime factor of k^2 - 1.

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Magma

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A223707 Conjectured least number k such that prime(n) is the largest divisor of k^3 + 1, or 0 if there is no such k.

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