A223701 -id:A223701 - cp's OEIS frontend

A181449 Numbers k such that 7 is the largest prime factor of k^2 - 1.

6, 8, 13, 15, 29, 41, 55, 71, 97, 99, 127, 244, 251, 449, 4801, 8749

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 7.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(4) = 8749; primepi(7) = 4.

Row 4 of A223701.

Cf. A076605, A175607, A181447, A181448, A181450-A181470, A181568.

[ n: n in [2..9000] | m eq 7 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011

Select[Range[9000], FactorInteger[#^2-1][[-1, 1]]==7&]

is(n)=n=n^2-1; forprime(p=2, 5, n/=p^valuation(n, p)); n>1 && 7^valuation(n, 7)==n \\ Charles R Greathouse IV, Jul 01 2013

A181450 Numbers k such that 11 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

10, 21, 23, 34, 43, 65, 76, 89, 109, 111, 197, 199, 241, 351, 485, 769, 881, 1079, 6049, 19601

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 11.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(5) = 19601; primepi(11) = 5.

Row 5 of A223701.

Cf. A076605, A175607, A181447-A181449, A181451-A181470, A181568.

[ n: n in [2..20000] | m eq 11 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 18 2011

Select[Range[20000], FactorInteger[#^2-1][[-1, 1]]==11&]

is(n)=n=n^2-1; forprime(p=2, 7, n/=p^valuation(n, p)); n>1 && 11^valuation(n, 11)==n \\ Charles R Greathouse IV, Jul 01 2013

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

Artur Jasinski, Oct 21 2010

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.

Row 6 of A223701.

Cf. A076605, A175607, A181447-A181450, A181452-A181470, A181568.

[ 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

Select[Range[250000], FactorInteger[#^2-1][[-1, 1]]==13&]

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

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

Andrew Howroyd, Dec 21 2024

nonn

See A175607 and A223701 for additional information.

Row sums of A223701.

Cf. A175607.

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

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

T. D. Noe, Apr 03 2013

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.

			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}

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

Cf. A223701-A223707 (related sequences).

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

T. D. Noe, Apr 03 2013

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.

			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}.

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

Cf. A223701-A223707 (related sequences).

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

A223705 Least number k such that prime(n) is the largest divisor of k^2 + 1, or 0 if there is no such k.

Original entry on oeis.org

1, 0, 2, 0, 0, 5, 4, 0, 0, 12, 0, 6, 9, 0, 0, 23, 0, 11, 0, 0, 27, 0, 0, 34, 22, 10, 0, 0, 33, 15, 0, 0, 37, 0, 44, 0, 28, 0, 0, 80, 0, 19, 0, 81, 14, 0, 0, 0, 0, 107, 89, 0, 64, 0, 16, 0, 82, 0, 60, 53, 0, 138, 0, 0, 25, 114, 0, 148, 0, 136, 42, 0, 0, 104, 0, 0

Offset: 1

T. D. Noe, Apr 03 2013

nonn

Note that a(n) = 0 for prime(n) = 3 (mod 4). If the zeros are removed, A002314 (with 1 prepended) and A177979 are produced.

Cf. A223701-A223707 (related sequences).

nn = 100; t = Table[0, {nn}]; Do[If[Mod[Prime[n], 4] == 3, t[[n]] = -1], {n, nn}]; n = 0; While[Times @@ t == 0, n++; s = FactorInteger[n^2 + 1][[-1, 1]]; p = PrimePi[s]; If[p <= nn && t[[p]] == 0, t[[p]] = n]]; Do[If[Mod[Prime[n], 4] == 3, t[[n]] = 0], {n, nn}]; t

A223706 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

0, 0, 0, 2, 0, 3, 18, 7, 0, 30, 5, 10, 0, 6, 6205, 0, 178, 13, 29, 0, 8, 23, 0, 0, 35, 102, 46, 429, 45, 0, 19, 263, 0, 42, 7600, 32, 12, 58, 1083163, 520, 0, 48, 880129, 84, 5320, 92, 14, 39, 13848, 94, 4195, 718, 15, 21085, 772, 502068, 5381, 28, 116, 5579256

Offset: 1

T. D. Noe, Apr 03 2013

nonn

We allowed k to vary up to 10^7.

Cf. A223701-A223707 (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

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

Andrew Howroyd, Dec 22 2024

See A181471 and A223701 for additional information.

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.)

Partial sums of A181471.

Cf. A175607, A223701, A379344.

cp's OEIS Frontend

A181449 Numbers k such that 7 is the largest prime factor of k^2 - 1.

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A181450 Numbers k such that 11 is the largest prime factor of k^2 - 1.

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

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

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

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

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A223705 Least number k such that prime(n) is the largest divisor of k^2 + 1, or 0 if there is no such k.

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A223706 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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A379345 Number of integers of the form k^2 - 1 whose greatest prime factor is at most prime(n).

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