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.

%I A223702 #11 Dec 22 2024 15:56:04
%S A223702 1,2,3,7,5,8,18,57,239,4,13,21,38,47,268,12,17,41,70,99,157,307,6,31,
%T A223702 43,68,117,191,302,327,882,18543,9,32,73,132,278,378,829,993,2943,23,
%U A223702 30,83,182,242,401,447,606,931,1143,1772,6118,34208,44179,85353,485298
%N 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.
%C A223702 Note that primes of the form 4x+3 are not divisors.
%H A223702 Andrew Howroyd, <a href="/A223702/b223702.txt">Table of n, a(n) for n = 1..811</a> (first 22 rows for primes up to 197)
%H A223702 Florian Luca, <a href="http://www.emis.de/journals/AMI/2004/acta2004-luca.pdf">Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1</a>, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
%H A223702 Filip Najman, <a href="http://web.math.hr/~fnajman/smooth.pdf">Smooth values of some quadratic polynomials</a>, Glasnik Matematicki Series III 45 (2010), pp. 347-355
%H A223702 Filip Najman, <a href="https://web.math.pmf.unizg.hr/~fnajman/publications.html">List of Publications Page</a> (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]
%e A223702 Irregular triangle:
%e A223702    p | {k}
%e A223702 -----+---------------------------------
%e A223702    2 | {1},
%e A223702    5 | {2, 3, 7},
%e A223702   13 | {5, 8, 18, 57, 239},
%e A223702   17 | {4, 13, 21, 38, 47, 268},
%e A223702   29 | {12, 17, 41, 70, 99, 157, 307},
%e A223702   37 | {6, 31, 43, 68, 117, 191, 302, 327, 882, 18543},
%e A223702   41 | {9, 32, 73, 132, 278, 378, 829, 993, 2943}
%e A223702   ...
%t A223702 t = Table[FactorInteger[n^2 + 1][[-1,1]], {n, 10^5}]; Table[Flatten[Position[t, Prime[n]]], {n, 13}]
%Y A223702 Cf. A002313, A014442, A177979 (first terms), A185389 (last terms), A223705, A285283, A379346 (row lengths), A379347 (row sums).
%Y A223702 Cf. A285282, A285523.
%Y A223702 Cf. A223701, A223703, A223704 (related tables).
%K A223702 nonn,tabf
%O A223702 1,2
%A A223702 _T. D. Noe_, Apr 03 2013
%E A223702 Definition amended by _Andrew Howroyd_, Dec 22 2024