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A233401 Numbers k such that k^3 - b2 is a triangular number (A000217), where b2 is the largest square less than k^3.

%I A233401 #21 Nov 14 2024 10:23:38
%S A233401 1,4,8,21,37,40,56,112,113,204,280,445,481,560,688,709,1933,1945,3601,
%T A233401 3805,3861,4156,4333,4365,7096,8408,8516,11064,12688,13609,13945,
%U A233401 16501,17080,18901,21464,23125,27244,27364,28141,45228,45549,58321,60061,66245,70585,78688
%N A233401 Numbers k such that k^3 - b2 is a triangular number (A000217), where b2 is the largest square less than k^3.
%C A233401 The cubes k^3 begin: 1, 64, 512, 9261, 50653, 64000, 175616, 1404928, ...
%C A233401 The squares b2 begin: 0, 49, 484, 9216, 50625, 63504, 175561, 1404225, ...
%C A233401 Their square roots are 0, 7, 22, 96, 225, 252, 419, 1185, 1201, 2913, 4685, 9387, ...
%o A233401 (Python)
%o A233401 from math import isqrt
%o A233401 def isTriangular(a):
%o A233401   a+=a
%o A233401   sr = isqrt(a)
%o A233401   return (a==sr*(sr+1))
%o A233401 for n in range(1,79999):
%o A233401   n3 = n*n*n
%o A233401   b = isqrt(n3)
%o A233401   if b*b==n3: b-=1
%o A233401   if isTriangular(n3-b*b):  print(n, end=', ')
%o A233401 (PARI) f(k) = if (issquare(k), sqrtint(k-1)^2, sqrtint(k)^2); \\ adapted from A048760
%o A233401 isok(k) = my(b2 = sqrtint(k^3-1)^2); (k^3-b2) && ispolygonal(k^3-b2, 3); \\ _Michel Marcus_, Jan 26 2019
%Y A233401 Cf. A000217, A000290, A000578, A048760, A233400.
%K A233401 nonn
%O A233401 1,2
%A A233401 _Alex Ratushnyak_, Dec 09 2013