A185389 -id:A185389 - cp's OEIS frontend

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.

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

A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 4, 9, 15, 22, 32, 41, 57, 74, 94, 120, 156, 192, 232, 278, 325, 381, 448, 521, 607, 704, 811

Offset: 1

Tomohiro Yamada, Apr 16 2017

In other words, x^2 + 1 is A002313(n)-smooth.
Størmer shows that the number of such integers is finite for any n.
a(n) <= 3^n - 2^n follows from Størmer's argument.
a(n) <= (2^n-1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
Luca 2004 determines all integers x such that x^2 + 1 is 100-smooth, which is pushed to 200 by Najman 2010.

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57--69.
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, Glas. Mat. 45 (2010), 347--355. Tables are available in the author's Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197).
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.

Equivalents for x(x+1): A145604.

Cf. A002313, A014442, A185389, A223702, A285282, A379346 (first differences).

a(13)-a(22) added by Andrew Howroyd, Dec 22 2024

A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

Original entry on oeis.org

2, 10, 108, 235, 1201, 390050, 314766, 4035, 364384, 50411, 25955045, 5254864, 236558593, 16958526, 20388056, 177544434, 492981885, 2275400230, 256347346, 384902923486, 324850200677887

Offset: 1

Charles R Greathouse IV, Feb 21 2011

nonn

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

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, Home Page (gives all 537 numbers x such that x^2-2 has no prime factor greater than 199)

Cf. A242488, A379348.

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1).

a(21) added by Andrew Howroyd, Dec 22 2024

A379346 Number of 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, 3, 5, 6, 7, 10, 9, 16, 17, 20, 26, 36, 36, 40, 46, 47, 56, 67, 73, 86, 97, 107

Offset: 1

Andrew Howroyd, Dec 22 2024

See A223702 for additional information.

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.

Row lengths of A223702.
First differences of A285283.
Cf. A002313, A177979, A185389.

A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

22, 140, 707, 21362, 4991, 1306066, 137965, 2294636, 31768298, 1557652, 340064590, 38439662, 105080665, 273502688, 543164542, 9575480365630, 391890109484, 14629598023, 80849485336, 1241646894380

Offset: 1

Charles R Greathouse IV, Feb 21 2011

nonn

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

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, Home Page (gives all 914 numbers x such that x^2+2 has no prime factor greater than 193)

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1), A185396 (x^2 - 2).

A249132 Smallest noncomposite k such that prime(n) is the largest prime factor of k^2+1, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 2, 0, 0, 5, 13, 0, 0, 17, 0, 31, 73, 0, 0, 23, 0, 11, 0, 0, 173, 0, 0, 233, 463, 293, 0, 0, 251, 919, 0, 0, 37, 0, 193, 0, 443, 0, 0, 599, 0, 19, 0, 467, 211, 0, 0, 0, 0, 107, 89, 0, 659, 0, 241, 0, 2503, 0, 337, 53, 0, 3671, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Oct 22 2014

nonn

a(A080148(m)) = 0. - Joerg Arndt, Oct 22 2014

			a(1)=1 is in this sequence because 1 is in A008578 and the largest prime factor of 1^2+1 = 2 is prime(1).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080148, A185389, A223702, A223705.

A249132:= proc(n) local p,i,k,a,b;
   p:= ithprime(n);
   if p mod 4 = 3 then return 0 fi;
   a:= numtheory:-msqrt(-1,p);
   if a < p/2 then b:= p-a
   else b:= a; a:= p-a
   fi;
   for i from 0 do
    for k in [a+i*p,b+i*p] do
      if isprime(k) and p = max(numtheory:-factorset(k^2+1)) then
        return(k)
      fi
    od
   od
end proc:
1,seq(A249132(n),n=2..100); # Robert Israel, Nov 10 2014

a249132[n_Integer] := Module[{t = Table[0, {n}], k, s, p}, Do[If[Mod[Prime[k], 4] == 3, t[[k]] = -1], {k, n}]; k = 0; While[Times @@ t == 0, k++; s = FactorInteger[k^2 + 1][[-1, 1]]; p = PrimePi[s]; If[p <= n && t[[p]] == 0 && ! CompositeQ[k], t[[p]] = k]]; t /. -1 -> 0]; a249132[120] (* Michael De Vlieger, Nov 11 2014, adapted from A223702 *)

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.

cp's OEIS Frontend

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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A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

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A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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A379346 Number of 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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A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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A249132 Smallest noncomposite k such that prime(n) is the largest prime factor of k^2+1, or 0 if no such k exists.

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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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