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

15120, 712800, 3341520, 10533600, 23284800, 85503600, 147026880, 171097920, 302702400, 477338400, 2058376320, 2633510880, 4204418400, 7342876800, 9673606800, 13035884400, 13734761040, 14895223200, 22388788800, 22647794400, 26108082000, 34183749600, 62246804400, 89169141600

Offset: 1

Author

Giedrius Alkauskas, Jul 18 2025

Keywords

• nonn

Comments

a(n) is divisible by 720.

Subsequence of A072389 (with two consecutive instead of four).

Integers k with five consecutive integers in the set {d+k/d : d|k} seem not to exist.

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

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

Examples

			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.
		

Links

• Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}
• David A. Corneth, 360 x 360 pixels image where white pixels with coordinate (k, m) have 720 | (k * (k + 1) * m * (m + 1))
• David A. Corneth, PARI program

Crossrefs

Cf. A002496, A027862, A072389, A335572, A386302.

Programs

Maple

M:=2*10^10:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) and issqr(3*F^2-2*G^2+6) then
       x:=(F+G-1)/2:
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

Extensions

More terms from David A. Corneth, Jul 19 2025