cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A225729 Numbers n such that n < d(n)^(21/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 210, 216, 240, 252, 264, 270, 280, 288, 300, 312, 330, 336, 360, 396
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^10 < d(n)^21. The last odd number is a(10) = 15.

Links

Crossrefs

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).
Cf. A225730-A225738.

Programs

  • Mathematica
    t = {}; Do[If[n < DivisorSigma[0, n]^(21/10), AppendTo[t, n]], {n, 10^4}]; t

A353448 Number w is in this sequence if every frame w X h, w >= h >= 3, contains more distinct distance quadrilaterals with corners interior to the 4 sides with concurrent diagonals, i.e., both ascending or both descending, than non-concurrent diagonals, or equivalently A353450(w,h) >= A353449(w,h).

Original entry on oeis.org

3, 4, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 29, 31, 33, 34, 37, 41, 43, 45, 46, 49, 55, 57, 61, 67, 73, 79, 81, 85, 89, 91, 97, 109, 113, 121, 127, 133, 141, 145, 151, 157, 161, 169, 181, 193, 201, 205, 209, 211, 217, 221, 225, 241, 253, 261, 265, 271
Offset: 1

Views

Author

Rainer Rosenthal, May 22 2022

Keywords

Comments

It is conjectured that this sequence is a subsequence A160007 except for the small terms <= 46 (verified for all w < 661). The example section depicts the way this sequence matches the irregular pattern of A160007. Numbers w > 100 were computed by Hugo Pfoertner.
w = 617 is one of the rare occurrences of remainder 17 mod 60 in A160007, and it is also in this sequence. One might suspect that there would be less and less hits, but the time-consuming computation successfully countered the intuition.

Examples

			.
w = 5 is in this sequence:
.
                        5 | . . . C .
   4 | . . C . .        4 | .       .    w = 5 is in this sequence because all
   3 | .       B        3 | .       B    quadrilaterals in (5,4) and (5,5)
   2 | D       .        2 | D       .    shown in the example section of A353450
   1 | . A . . .        1 | . A . . .    have concurrent diagonals.
   y /----------        y /----------
     x 1 2 3 4 5          x 1 2 3 4 5
.
w = 6 is *not* in this sequence:
.
   4 | . . . C . .         w = 6 is not in this sequence because of the single
   3 | D         .         quadrilateral in (6,4) shown in the example section
   2 | .         B         of A353449. Diagonal AC is rising while diagonal DB
   1 | . A . . . .         is falling (non-concurrent diagonals).
   y /------------         There is no (6,4) quadrilateral with all distances
     x 1 2 3 4 5 6         distinct and with concurrent diagonals!
.
            123456789012345678901234567890123456789012345678901234567890
    1 -  60   ::: x  :: x:::: x x : x   x x ::  x   x x ::  x     x x
   61 - 120 x     x   . x     x x   x   x x     x   . . .   x   x .
  121 - 180 x     x     x     . x   x     x     x   x .     x     . .
  181 - 240 x     .     x   . . x   x   x x     x   x . x   .     .
  241 - 300 x     .     x     . x   x     x .   .   x .     x     .
  301 - 360 x   . . .   x     . x   x     .     x   . .     . . . .
  361 - 420 x   . . .   .     x .   x     x .   x   x .     x     . .
  421 - 480 x     .     x     . x   .   . .     x   . x .   x     . .
  481 - 540 x     .   . .   . . .   x     .     .   x .     x   . .
  541 - 600 x   . .   . .     . x   .     . .   x   . .     x     .
  601 - 660 x     . .   x   x . .   x     x     .   x . .   x .   .
.
Legend:
  "x" marks numbers w belonging to this sequence and to A160007.
  ":" marks numbers w belonging to this sequence only.
  "." marks numbers w belonging to A160007 only.

Links

Crossrefs

Cf. A160007, A353532 ("all tetrapods"), A353449 ("unisense"), A353450 ("contrasense").
Cf. A225730 (has many terms in common when 1 is added, see also comparison plot).
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