A215965 Numbers n such that the absolute value of the difference between the sum of the distinct prime factors of n^2 + 1 that are congruent to 1 mod 8 and the sum of the distinct prime factors of n^2 + 1 that are congruent to 5 mod 8 is a square. There must be at least one prime factor of each type.

21, 30, 32, 47, 72, 191, 268, 274, 313, 327, 401, 469, 500, 526, 606, 775, 785, 788, 820, 889, 891, 919, 948, 1046, 1250, 1382, 1428, 1441, 1560, 1636, 1696, 1714, 1772, 1806, 1834, 2018, 2041, 2348, 2584, 2610, 2641, 2782, 3144, 3357, 3359, 3488, 3740, 3769

Offset

1,1

Comments

The corresponding squares are : 4, 36, 36, 1, 49, 49, 1, 196, 9, 25, 49, 900, 36, 484, 25, 1764, 49, 256, 1089, 169, 1156, 25, 0, 5476, 100, 81, 49, 529, 0, 16, ... and the values n for which this sequence equals 0 are in A215950.

Links

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

Example

2018 is in the sequence because 2018^2 + 1 = 4072325 = 5^2*29*41*137 and (137+41) - (5+29) = 144 is a square, where {41, 137} == 1 mod 8 and {5, 29} ==5 mod 8.

Maple

with(numtheory):for n from 1 to 1000 do:x:=factorset(n^2+1):n1:=nops(x):s1:=0:s3:=0:for m from 1 to n1 do: if irem(x[m], 8)=1 then s1:=s1+x[m]:else if irem(x[m], 8)=5 then s3:=s3+x[m]:else fi:fi:od:x:=abs(s1-s3):y:=sqrt(x):if s1>0 and s3>0 and y=floor(y)  then printf(`%d, `, n):else fi:od:

Crossrefs

Cf. A002522, A215950, A215963.

Sequence in context: A039372 A043195 A043975 * A225653 A363609 A336357

Adjacent sequences: A215962 A215963 A215964 * A215966 A215967 A215968

Keyword

nonn

Author

Michel Lagneau, Aug 29 2012

Extensions

Definition clarified by Robert Israel, Sep 27 2018

Status

approved