A379344 -id:A379344 - cp's OEIS frontend

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

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}]

A379347 a(n) is the sum of all integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 12, 327, 391, 703, 20510, 5667, 661016, 507004, 644098, 24977604, 38394505, 2621510449, 465558141, 624692559, 63435958, 507041846, 8133206945, 70119049516045, 45102364892, 49035127231, 154823547391

Offset: 1

Andrew Howroyd, Dec 22 2024

See A223702 for additional information.

			a(2) = 12 = 2 + 3 + 7. The corresponding values for k^2 + 1 are 5, 10 and 50 each of whose greatest prime factor is 5 = A002313(2).

Row sums of A223702.

Cf. A002313, A185389, A379344.

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

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

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A379347 a(n) is the sum of all integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

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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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