login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A113688 Isolated semiprimes in the semiprime spiral. 9

%I

%S 65,74,249,295,309,355,422,511,545,667,669,758,926,943,979,998,1099,

%T 1167,1186,1322,1457,1469,1561,1585,1658,1711,1774,1779,1835,1891,

%U 1959,1961,1963,2021,2038,2066,2155,2186,2191,2206,2271,2329,2342

%N Isolated semiprimes in the semiprime spiral.

%C Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. Analogous to Ulam's marking the primes in the spiral and discovering unexpectedly many connected diagonals, we construct a semiprime spiral by marking the semiprimes (A001358). Each integer has 8 adjacent integers in the spiral, horizontally, vertically and diagonally. Curious extended clumps coagulate, slightly denser towards the origin, of semiprimes connected by adjacency. This sequence lists the isolated semiprimes in the semiprime spiral, namely those semiprimes none of whose adjacent integers in the spiral are semiprimes. A113689 gives an enumeration of the number of semiprimes in clumps of size >1 through n^2.

%D S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.

%H Michael De Vlieger, <a href="/A113688/b113688.txt">Table of n, a(n) for n = 1..2000</a>

%H Alois P. Heinz, <a href="/A113688/a113688.png">Plot of semiprime spiral</a>, containing all semiprimes <= 10000. Isolated semiprimes are colored red.

%H M. Stein and S. M. Ulam, <a href="http://www.jstor.org/stable/2314055">An Observation on the Distribution of Primes</a>, Amer. Math. Monthly 74, 43-44, 1967.

%H M. Stein and S. M. Ulam and M. B. Wells, <a href="http://www.jstor.org/stable/2312588">A Visual Display of Some Properties of the Distribution of Primes</a>, Amer. Math. Monthly 71, 516-520, 1964.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.

%e Spiral example:

%e ......................

%e ... 17 16 15 14 13 ...

%e ... 18 5 4 3 12 ...

%e ... 19 6 1 2 11 ...

%e ... 20 7 8 9 10 ...

%e ... 21 22 23 24 25 ...

%e ......................

%e From _Michael De Vlieger_, Dec 22 2015: (Start)

%e Spiral including n <= 121 showing only semiprimes, the isolated semiprimes appear in parentheses:

%e . . . . . . 95 94 93 . 91

%e . (65) . . 62 . . . 58 57 .

%e . . . . 35 34 33 . . . .

%e . . 38 . . 15 14 . . 55 .

%e . . 39 . . 4 . . . . 87

%e 106 69 . . 6 . . . . . 86

%e . . . . . . 9 10 . . 85

%e . . . 21 22 . . 25 26 51 .

%e . . . . . 46 . . 49 . .

%e . . (74) . . 77 . . . . 82

%e 111 . . . 115 . . 118 119 . 121

%e (End)

%t spiral[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; f[w_] := Block[{d = Dimensions@ w, t, g}, t = Reap[Do[Sow@ Take[#[[k]], {2, First@ d - 1}], {k, 2, Last@ d - 1}]][[-1, 1]] &@ w; g[n_] := If[n != 0, Total@ Join[Take[w[[Last@ # - 1]], {First@ # - 1, First@ # + 1}], {First@ #, Last@ #} &@ Take[w[[Last@ #]], {First@ # - 1, First@ # + 1}], Take[w[[Last@ # + 1]], {First@ # - 1, First@# + 1}]] &@(Reverse@ First@ Position[t, n] + {1, 1}) == 0, False]; Select[Union@ Flatten@ t, g@ # &]]; t = spiral@ 26 /. n_ /; PrimeOmega@ n != 2 -> 0; f@ t (* _Michael De Vlieger_, Dec 21 2015, Version 10 *)

%Y Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Nov 05 2005

%E Corrected and extended by _Alois P. Heinz_, Jan 02 2011

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 24 22:41 EST 2018. Contains 299627 sequences. (Running on oeis4.)