A379705 a(n) is the least integer k > n such that integers p, q exist for which n, p, k are in arithmetic and n, q, k are in geometric progression.

9, 8, 27, 16, 45, 24, 63, 18, 25, 40, 99, 48, 117, 56, 135, 36, 153, 32, 171, 80, 189, 88, 207, 54, 49, 104, 75, 112, 261, 120, 279, 50, 297, 136, 315, 64, 333, 152, 351, 90, 369, 168, 387, 176, 125, 184, 423, 108, 81, 72, 459, 208, 477, 96, 495, 126, 513, 232

Offset: 1

Author

Felix Huber, Jan 07 2025

Keywords

• nonn

Examples

			a(9) = 25 because 9, 17, 25 are in arithmetic progression (common difference = 8) and 9, +-15, 25 are in geometric progression (common ratio = +-5/3) and there is no other integer k with 9 < k < 25 such that integers p and q exist for which 9, p, k are in arithmetic and 9, q, k are in geometric progression.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Wikipedia, Arithmetic Progression
• Wikipedia, Geometric Progression

Crossrefs

Cf. A000188, A008833, A057918, A072905, A379663.

Programs

Maple

A379705:=proc(n)
   local d;
   d:=expand(NumberTheory:-LargestNthPower(n,2));
   if is(n*(1+(d+1)^2/d^2),even) then
      n*(d+1)^2/d^2
   else
      n*(d+2)^2/d^2
   fi;
end proc;
seq(A379705(n),n=1..58);

Formula

a(n) = n/A008833(n)*(A000188(n) + k)^2, where k = 1 if n*(1+(A000188(n)+1)^2/A008833(n)) is even or k = 2 else.

a(n) = A072905(n) if n*(1+(A000188(n)+1)^2/A008833(n)) is even.