This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348169 #62 Dec 19 2024 11:45:36 %S A348169 3,12,18,27,30,42,48,72,75,77,98,108,120,147,154,162,168,192,243,255, %T A348169 260,264,270,272,273,285,288,297,300,308,338,363,378,392,432,450,462, %U A348169 480,490,494,507,510,513,588,616,630,648,672,675,693,702,714,722,750,754,768,798 %N A348169 Positive integers which can be represented as A*(x^2 + y^2 + z^2) = B*(x*y + x*z + y*z) with positive integers x, y, z, A, B and gcd(A,B)=1. %C A348169 The sequence represents a generalization of cases A033428 (k=1), A347960 (k=2), A347969 (k=5) with all possible k given by A331605. Instead of integer k, it utilizes the ratio B/A. %H A348169 Chai Wah Wu, <a href="/A348169/b348169.txt">Table of n, a(n) for n = 1..10000</a> %H A348169 Alexander Kritov, <a href="/A348169/a348169_2.c.txt">Source code</a> %e A348169 a(6)=42: the quintuple (x,y,z) A,B is 1,2,4 (2,3) because 42 = 2*(1^2 + 2^2 + 4^2) = 3*(1*4 + 1*2 + 2*4). %e A348169 a(n) (x,y,z) A, B %e A348169 3 (1,1,1) 1, 1 %e A348169 12 (2,2,2) 1, 1 %e A348169 18 (1,1,4) 1, 2 %e A348169 27 (3,3,3) 1, 1 %e A348169 30 (1,1,2) 5, 6 %e A348169 42 (1,2,4) 2, 3 %e A348169 48 (4,4,4) 1, 1 %e A348169 72 (1,2,2) 8, 9 [also (2,2,8) 1, 2] %e A348169 75 (5,5,5) 1, 1 %e A348169 77 (1,1,3) 7, 11 %e A348169 98 (1,4,9) 1, 2 %e A348169 108 (6,6,6) 1, 1 %e A348169 120 (2,2,4) 5, 6 %e A348169 147 (7,7,7) 1, 1 %e A348169 154 (1,2,3) 11, 14 %e A348169 162 (3,3,12) 1, 2 %e A348169 168 (2,4,8) 2, 3 %e A348169 192 (8,8,8) 1, 1 %e A348169 243 (9,9,9) 1, 1 %e A348169 255 (1,1,7) 5, 17 %e A348169 260 (2,5,6) 4, 5 %e A348169 264 (1,4,4) 8, 11 %e A348169 270 (2,5,5) 5, 6 %e A348169 272 (2,2,3) 16, 17 %e A348169 288 (4,4,2) 8, 9 [also (4,4,16) 1, 2] %o A348169 (C) /* See links */ %o A348169 (Python) %o A348169 from itertools import islice, count %o A348169 from math import gcd %o A348169 from sympy import divisors, integer_nthroot %o A348169 def A348169(): # generator of terms %o A348169 for n in count(1): %o A348169 for d in divisors(n,generator=False): %o A348169 x, x2 = 1, 1 %o A348169 while 3*x2 <= d: %o A348169 y, y2 = x, x2 %o A348169 z2 = d-x2-y2 %o A348169 while z2 >= y2: %o A348169 z, w = integer_nthroot(z2,2) %o A348169 if w: %o A348169 A = n//d %o A348169 B, u = divmod(n,x*(y+z)+y*z) %o A348169 if u == 0 and gcd(A,B) == 1: %o A348169 yield n %o A348169 break %o A348169 y += 1 %o A348169 y2 += 2*y-1 %o A348169 z2 -= 2*y-1 %o A348169 else: %o A348169 x += 1 %o A348169 x2 += 2*x-1 %o A348169 continue %o A348169 break %o A348169 else: %o A348169 continue %o A348169 break %o A348169 A348169_list = list(islice(A348169(),57)) # _Chai Wah Wu_, Nov 26 2021 %Y A348169 Cf. A331605, A347969, A347960. %Y A348169 The sequence contains A033428 (A=B=1), A347969 (B=2*A), A347960 (B=5*A). %K A348169 nonn %O A348169 1,1 %A A348169 _Alexander Kritov_, Oct 04 2021