A113653 - OEIS

%I #45 Apr 14 2023 11:21:13

%S 6,51,69,82,91,183,194,221,249,265,287,289,309,314,319,323,355,371,

%T 403,417,437,469,478,511,517,519,533,579,589,649,681,689,731,749,758,

%U 807,838,849,926,943,951,961,965,979,1011,1018,1037,1055,1057,1067,1077,1099,1126,1145,1149,1154,1159

%N Isolated semiprimes in the hexagonal spiral.

%C Isolated semiprimes in the hexagonal spiral of A003215 and A001399, which is centered on 0. Of course such a spiral can be constructed beginning with any integer. Centering on 0 gives the interesting partition and multigraph equalities of A001399.

%C Integers in A001358 which are not adjacent in any of six directions to any other integer in A001358 when arranged in the hexagonal spiral.

%C An analog of  A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld].

%C Unfortunately the original submission (which has been preserved as the "dead" sequence A335704) omitted the number 44 from the spiral, which has caused an enormous amount of trouble. - _N. J. A. Sloane_, Jun 27 2020

%D Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

%H P. Abbott, <a href="http://forums.wolfram.com/mathgroup/archive/2005/May/msg00336.html">Re: Hexagonal Spiral</a>, <a href="http://groups-beta.google.com/group/comp.soft-sys.math.mathematica">(alt link)</a>, May 11, 2005

%H H. Bottomley, <a href="/A003215/a003215.gif">Spokes of a Hexagonal Spiral.</a>

%H R. J. Mathar, <a href="/A113653/a113653.txt">Maple program for A113653</a>

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

%e The spiral begins:

%e                 120-119-118-117-116-115-114

%e                  /                         \

%e               121  85--84--83-*82*-81--80 113

%e                /   /                     \   \

%e             122  86  56--55--54--53--52  79 112

%e              /   /   /                 \   \   \

%e           123  87  57  33--32--31--30 *51* 78 111

%e            /   /   /   /             \   \   \   \

%e         124  88  58  34  16--15--14  29  50  77 110

%e          /   /   /   /   /         \   \   \   \   \

%e       125  89  59  35  17   5---4  13  28  49  76 109

%e        /   /   /   /   /   /     \   \   \   \   \   \

%e     126  90  60  36  18  *6*  0   3  12  27  48  75 108

%e      /   /   /   /   /   /   /   /   /   /   /   /   /

%e   127 *91* 61  37  19   7   1---2  11  26  47  74 107 146

%e      \   \   \   \   \   \         /   /   /   /   /   /

%e     128  92  62  38  20   8---9--10  25  46  73 106 145

%e        \   \   \   \   \             /   /   /   /   /

%e       129  93  63  39  21--22--23--24  45  72 105 144

%e          \   \   \   \                 /   /   /   /

%e         130  94  64  40--41--42--43--44  71 104 143

%e            \   \   \                     /   /   /

%e           131  95  65--66--67--68-*69*-70 103 142

%e              \   \                         /   /

%e             132  96--97--98--99-100-101-102 141

%e                \                             /

%e               133-134-135-136-137-138-139-140

%Y Cf. A001358, A001399, A113688.

%Y For the sequence of isolated primes see A335916.

%Y Related sequences:

%Y A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105).

%Y A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106).

%Y A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107).

%Y A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108).

%Y A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109).

%Y A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215).

%K nonn

%O 1,1

%A _Jonathan Vos Post_, Jan 16 2006

%E Corrected and edited by _N. J. A. Sloane_, Jun 27 2020. Thanks to Jeffrey K. Aronson for pointing out the error in the original submission.

%E Terms a(4) onwards corrected by _R. J. Mathar_, Jun 29 2020