A132067 Composite integers m where d_{k+2} + d_k < 2*d_{k+1} for at least one k (1<=k<=A000005(m)-2), where d_k is the k-th positive divisor of m.

20, 30, 35, 40, 42, 56, 60, 63, 70, 72, 77, 80, 84, 88, 90, 99, 100, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 150, 154, 156, 160, 165, 168, 175, 176, 180, 182, 187, 189, 195, 198, 200, 204, 208, 209, 210, 216, 220, 221, 224, 238, 240, 245, 247

Offset

1,1

Comments

In other words, the sequence contains those positive integers m where the difference (d_{k+1} - d_k) between some pair of consecutive positive divisors of m is greater than the difference (d_{k+2} - d_{k+1}) between the next pair of consecutive divisors of m.

Links

Miles Englezou, Table of n, a(n) for n = 1..10000

Example

The positive divisors of 20 are 1,2,4,5,10,20. d_2 + d_4 = 2 + 5 is < 2 * d_3 = 2 * 4. So 20 is a term.

Mathematica

f[n_] := Block[{d}, d = Divisors[n]; d - Prepend[Most[d], 0]]; Flatten[Position[OrderedQ /@ Array[f, 260], False]] (* Ray Chandler, Nov 01 2007 *)
a = {}; For[n = 1, n < 1000, n++, c = 0; For[j = 1, j < Length[Divisors[n]] - 1, j++, If[Divisors[n][[j]] + Divisors[n][[j + 2]] < 2*Divisors[n][[j + 1]], c = 1]]; If[c == 1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Oct 31 2007 *)

Prog

(PARI) isok(n) = my(D = divisors(n), A = apply(x -> D[x] - D[x-1], [2..#D])); if(vecprod(apply(x -> A[x] >= A[x-1], [2..#A])) <> 1, 1, 0) \\ Miles Englezou, Jan 06 2026

Crossrefs

Sequence in context: A107714 A029721 A224400 * A072989 A216603 A166730

Adjacent sequences: A132064 A132065 A132066 * A132068 A132069 A132070

Keyword

nonn

Author

Leroy Quet, Oct 30 2007

Extensions

Extended by Ray Chandler and Stefan Steinerberger, Nov 01 2007

Status

approved