Tim Cieplowski - cp's OEIS frontend

A262767 Minimum perimeter of a rectangle with area n and integer sides.

4, 6, 8, 8, 12, 10, 16, 12, 12, 14, 24, 14, 28, 18, 16, 16, 36, 18, 40, 18, 20, 26, 48, 20, 20, 30, 24, 22, 60, 22, 64, 24, 28, 38, 24, 24, 76, 42, 32, 26, 84, 26, 88, 30, 28, 50, 96, 28, 28, 30, 40, 34, 108, 30, 32, 30, 44, 62, 120, 32

Offset: 1

Tim Cieplowski, Sep 30 2015

a(n) >= A027709(n) = 2*ceiling(2*sqrt(n)). - Dmitry Kamenetsky, Feb 27 2017

			Since 2 * (2 + 3) < 2 * (1+6), a(6) = 10.

Cf. A063655 (semiperimeter).

Two-dimensional equivalent of A075777.

f[n_] := Block[{w = Round@ Sqrt@ n}, While[Mod[n, w] != 0, w--]; 2 (w + Round[n/w])]; Array[f, {60}] (* Michael De Vlieger, Oct 01 2015 *)

a(n) = {local(d); d=divisors(n); 2*(d[(length(d)+1)\2] + d[length(d)\2+1])}
vector(50, n, a(n)) \\ Altug Alkan, Oct 16 2015

def perimeter(area):
    width = round(area ** (1/2))
    while area % width != 0:
        width -= 1
    return 2*(width + round(area/width))

a(n) = 2*A063655(n). - Michel Marcus, Oct 01 2015

A253620 Maximum number of segments in nonintersecting increasing path on n X n hexagonal (isogonal) grid.

Original entry on oeis.org

0, 3, 6, 10, 14, 19, 25, 30, 36

Offset: 1

Tim Cieplowski, Jan 06 2015

The path cannot intersect itself, not even on single points. "Increasing" means that the (Euclidean) length of each segment must be strictly greater than that of the previous one.

The analogous sequence for a triangular (isogonal) grid seems to satisfy a(n) = 2n+1, with 2^(n-2) such paths up to isomorphism.

			An example for a(4) = 10
       .   .   .   .
    09   .   .   .   .
  01   .   .   .   .   .
00  07   .   .   .   .  10
  02  05   .   .   .  08
     .   .   .   .  06
      03   .   .  04

Tim Cieplowski, Illustration of first few terms
Gordon Hamilton, $1,000,000 Unsolved Problem for Grade 8 (2011)

Cf. A226595.

A227050 Number of essentially different ways of arranging numbers 1 through 2n around a circle so that the sum and absolute difference of each pair of adjacent numbers are prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 4, 88, 0, 976, 22277, 22365, 376002, 3172018, 5821944, 10222624, 424452210, 6129894510, 38164752224

Offset: 1

Tim Cieplowski, Jun 29 2013

See a similar problem, but for the set of numbers {0 through (n-1)}. - Stanislav Sykora, May 30 2014

			For n = 6 the a(6) = 2 solutions are (1, 4, 9, 2, 5, 12, 7, 10, 3, 8, 11, 6) and (1, 6, 11, 8, 3, 10, 7, 4, 9, 2, 5, 12) because abs(1 - 4) = 3 and 1 + 4 = 5 are prime, etc.

Gary Antonick, Numberplay: Bernardo Recamán’s Primes in a Circle Puzzle, Jun 17 2013.
Stanislav Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014; Table III.

Cf. similar sequences: A051252 (with sums of neighbors prime), A242527 (with sums of neighbors prime), A228626 (with differences of neighbors prime), A242528 (with sums and differences of neighbors prime).

A227050[n_] :=
Count[Map[lpf, Map[j1f, Permutations[Range[2,2 n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
lpf[x_] := Length[
   Join[Select[asf[x], ! PrimeQ[#] &],
    Select[Differences[x], ! PrimeQ[#] &]]];
asf[x_] := Module[{i}, Table[x[[i]] + x[[i + 1]], {i, Length[x] - 1}]];
Table[A227050[n], {n, 1, 6}]
(* OR, a less simple, but more efficient implementation. *)
A227050[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[PrimeQ[First[perm] - Last[perm]] &&
       PrimeQ[First[perm] + Last[perm]], ct++];
     Return[ct],
     opt = remain; lr = Length[remain];
     For[i = 1, i <= lr, i++,
      new = First[opt]; opt = Rest[opt];
      If[! (PrimeQ[Last[perm] - new] && PrimeQ[Last[perm] + new]),
       Continue[]];
      A227050[n, Join[perm, {new}],
       Complement[Range[2 n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Table[ct = 0; A227050[n, {1}, Range[2, 2 n]]/2, {n, 1, 10}]
(* Robert Price, Oct 22 2018 *)

a(15)-a(18) added by Tim Cieplowski, Jan 04 2015
a(19) from Fausto A. C. Cariboni, Jun 06 2017
a(20) from Bert Dobbelaere, Feb 15 2020

A226108 Primes remaining prime if all but two digits are deleted.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1117, 1171, 4111, 11113, 11117, 11119, 11131, 11171, 11173, 11197, 11311, 11317, 11719, 11731, 13171, 13711, 41113

Offset: 1

Tim Cieplowski, May 26 2013

Subsequence of A069488.

			For a(3)=137, all pairs of two digits (in their original order) 13, 17, and 37 are prime.

C. Caldwell, Truncatable primes, J. Recreational Math., 19:1 (1987) 30-33.

Tim Cieplowski, Table of n, a(n) for n = 1..2720
P. Ballew, Knockout Primes and a new notation, May 17, 2013
C. Caldwell, The Prime Glossary: Deletable Prime

Cf. A019546, A051362, A069488.

testQ[n_] := n > 9 && Catch[Block[{d = IntegerDigits@n}, Do[If[! PrimeQ[ d[[j]] + 10*d[[i]]], Throw@False], {j, 2, Length@d}, {i, j-1}]; True]]; Select[Prime@ Range[10^5], testQ] (* Giovanni Resta, May 28 2013 *)

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User: Tim Cieplowski

A262767 Minimum perimeter of a rectangle with area n and integer sides.

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A253620 Maximum number of segments in nonintersecting increasing path on n X n hexagonal (isogonal) grid.

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A227050 Number of essentially different ways of arranging numbers 1 through 2n around a circle so that the sum and absolute difference of each pair of adjacent numbers are prime.

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A226108 Primes remaining prime if all but two digits are deleted.

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Mathematica