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

1,1

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Wikipedia, Arithmetic Progression

Wikipedia, Geometric Progression

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.

Example

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.

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);

Crossrefs

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

Sequence in context: A264301 A251091 A159078 * A253090 A154226 A177187

Adjacent sequences: A379702 A379703 A379704 * A379706 A379707 A379708

Keyword

nonn,new

Author

Felix Huber, Jan 07 2025

Status

approved