A215965 - OEIS

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.

%I #16 Sep 27 2018 19:57:45

%S 21,30,32,47,72,191,268,274,313,327,401,469,500,526,606,775,785,788,

%T 820,889,891,919,948,1046,1250,1382,1428,1441,1560,1636,1696,1714,

%U 1772,1806,1834,2018,2041,2348,2584,2610,2641,2782,3144,3357,3359,3488,3740,3769

%N 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.

%C 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.

%H Robert Israel, <a href="/A215965/b215965.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%p 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:

%Y Cf. A002522, A215950, A215963.

%K nonn

%O 1,1

%A _Michel Lagneau_, Aug 29 2012

%E Definition clarified by _Robert Israel_, Sep 27 2018