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.
%I A113653 #46 Feb 16 2025 08:32:59 %S A113653 6,51,69,82,91,183,194,221,249,265,287,289,309,314,319,323,355,371, %T A113653 403,417,437,469,478,511,517,519,533,579,589,649,681,689,731,749,758, %U A113653 807,838,849,926,943,951,961,965,979,1011,1018,1037,1055,1057,1067,1077,1099,1126,1145,1149,1154,1159 %N A113653 Isolated semiprimes in the hexagonal spiral. %C A113653 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 A113653 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 A113653 An analog of A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld]. %C A113653 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 A113653 Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990. %H A113653 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 A113653 H. Bottomley, <a href="/A003215/a003215.gif">Spokes of a Hexagonal Spiral.</a> %H A113653 R. J. Mathar, <a href="/A113653/a113653.txt">Maple program for A113653</a> %H A113653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral.</a> %e A113653 The spiral begins: %e A113653 120-119-118-117-116-115-114 %e A113653 / \ %e A113653 121 85--84--83-*82*-81--80 113 %e A113653 / / \ \ %e A113653 122 86 56--55--54--53--52 79 112 %e A113653 / / / \ \ \ %e A113653 123 87 57 33--32--31--30 *51* 78 111 %e A113653 / / / / \ \ \ \ %e A113653 124 88 58 34 16--15--14 29 50 77 110 %e A113653 / / / / / \ \ \ \ \ %e A113653 125 89 59 35 17 5---4 13 28 49 76 109 %e A113653 / / / / / / \ \ \ \ \ \ %e A113653 126 90 60 36 18 *6* 0 3 12 27 48 75 108 %e A113653 / / / / / / / / / / / / / %e A113653 127 *91* 61 37 19 7 1---2 11 26 47 74 107 146 %e A113653 \ \ \ \ \ \ / / / / / / %e A113653 128 92 62 38 20 8---9--10 25 46 73 106 145 %e A113653 \ \ \ \ \ / / / / / %e A113653 129 93 63 39 21--22--23--24 45 72 105 144 %e A113653 \ \ \ \ / / / / %e A113653 130 94 64 40--41--42--43--44 71 104 143 %e A113653 \ \ \ / / / %e A113653 131 95 65--66--67--68-*69*-70 103 142 %e A113653 \ \ / / %e A113653 132 96--97--98--99-100-101-102 141 %e A113653 \ / %e A113653 133-134-135-136-137-138-139-140 %Y A113653 Cf. A001358, A001399, A113688. %Y A113653 For the sequence of isolated primes see A335916. %Y A113653 Related sequences: %Y A113653 A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105). %Y A113653 A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106). %Y A113653 A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107). %Y A113653 A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108). %Y A113653 A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109). %Y A113653 A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215). %K A113653 nonn %O A113653 1,1 %A A113653 _Jonathan Vos Post_, Jan 16 2006 %E A113653 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 A113653 Terms a(4) onwards corrected by _R. J. Mathar_, Jun 29 2020