A317518 - OEIS

%I #25 Dec 14 2021 01:45:56

%S 1,4,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,7560,

%T 10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,166320,

%U 221760,277200,332640,554400

%N Highly composite numbers k such that ceiling(sqrt(k))^2 - k is a square.

%C Up to 17297280 only the highly composite numbers 2, 6, 498960, 1081080, and 4324320 don't qualify. I have tested all up to that point.

%C Conjecture: this sequence is finite, with a(120) = 7675044034503567507122937600 as its final term. - _Jon E. Schoenfield_, Aug 12 2018

%H Michael De Vlieger, <a href="/A317518/b317518.txt">Table of n, a(n) for n = 1..120</a>

%H Michael De Vlieger, <a href="/A317518/a317518.txt">Concordance of A317518 and A002182</a> listing a(n) and positions of a(n) in A002182, along with the qualifying square roots.

%F Intersection of A002182 and A256173. - _Andrew Howroyd_, Aug 12 2018

%e 1, 4, and 36 are square. 24, 48, 120, 360, 840, 1680, and 5040 are all 1 less than a square.

%o (PARI) {my(r=0); for(k=1, 5e5, if(numdiv(k)>r, r=numdiv(k); if(issquare((sqrtint(k-1) + 1)^2 - k), print1(k, ", ")) ))} \\ _Andrew Howroyd_, Aug 12 2018

%Y Cf. A002182, A256173.

%K nonn

%O 1,2

%A _Vaughn R Tiffany_, Jul 30 2018