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A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

%I A386303 #41 Jul 22 2025 10:35:08
%S A386303 15120,712800,3341520,10533600,23284800,85503600,147026880,171097920,
%T A386303 302702400,477338400,2058376320,2633510880,4204418400,7342876800,
%U A386303 9673606800,13035884400,13734761040,14895223200,22388788800,22647794400,26108082000,34183749600,62246804400,89169141600
%N A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.
%C A386303 a(n) is divisible by 720.
%C A386303 Subsequence of A072389 (with two consecutive instead of four).
%C A386303 Integers k with five consecutive integers in the set {d+k/d : d|k} seem not to exist.
%C A386303 As terms must be of the form k * (k + 1) * m * (m + 1) and divisible by 720 we can restrict the search based on g = gcd(k * (k + 1), 720) which is at least 2. We must have (720 / g) | m * (m + 1). - _David A. Corneth_, Jul 19 2025
%C A386303 If q is the number of divisors of a(n) then the first of these four divisors is generally d[q/2 + 1] at least for nonsquares. For three consecutive integers (cf. A386302) there is the exception 180180. - _David A. Corneth_, Jul 20 2025
%H A386303 Giedrius Alkauskas, <a href="https://web.vu.lt/mif/g.alkauskas/math/consecutive-integers.pdf">Consecutive integers in the set S_n={d+n/d: d|n}</a>
%H A386303 David A. Corneth, <a href="/A386303/a386303.png">360 x 360 pixels image where white pixels with coordinate (k, m) have 720 | (k * (k + 1) * m * (m + 1))</a>
%H A386303 David A. Corneth, <a href="/A386303/a386303_1.gp.txt">PARI program</a>
%e A386303 a(1)=15120=M is a term of this sequence since 105, 108, 112, 120 are divisors of M, and 120+M/120=246, 112+M/112=247, 108+M/108=248, 105+M/105=249. It is the first term since no smaller such positive integer exists.
%p A386303 M:=2*10^10:
%p A386303 Ki:={}:
%p A386303 Vi:=floor(sqrt(2*M)):
%p A386303 Ski:=floor((19*M)^(1/4)/2):
%p A386303 for F from 1 to Vi-4 do
%p A386303   for y from 1 to min(floor((Vi-F)/2),Ski) do
%p A386303      G:=F+2*y+1:
%p A386303      if issqr(2*F^2-G^2+2) and issqr(3*F^2-2*G^2+6) then
%p A386303        x:=(F+G-1)/2:
%p A386303        n:=x*(x+1)*y*(y+1):
%p A386303        Ki:=Ki union {n}:
%p A386303      end if:
%p A386303   end do:
%p A386303 end do:
%p A386303 Ki;
%Y A386303 Cf. A002496, A027862, A072389, A335572, A386302.
%K A386303 nonn
%O A386303 1,1
%A A386303 _Giedrius Alkauskas_, Jul 18 2025
%E A386303 More terms from _David A. Corneth_, Jul 19 2025