A113653 - OEIS

A113653

Isolated semiprimes in the hexagonal spiral.

6, 51, 69, 82, 91, 183, 194, 221, 249, 265, 287, 289, 309, 314, 319, 323, 355, 371, 403, 417, 437, 469, 478, 511, 517, 519, 533, 579, 589, 649, 681, 689, 731, 749, 758, 807, 838, 849, 926, 943, 951, 961, 965, 979, 1011, 1018, 1037, 1055, 1057, 1067, 1077, 1099, 1126, 1145, 1149, 1154, 1159

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OFFSET

1,1

COMMENTS

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.

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

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

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..57.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005

H. Bottomley, Spokes of a Hexagonal Spiral.

R. J. Mathar, Maple program for A113653

Eric Weisstein's World of Mathematics, Prime Spiral.

EXAMPLE

The spiral begins:

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

/ \

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

/ / \ \

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

/ / / \ \ \

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

/ / / / \ \ \ \

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

/ / / / / \ \ \ \ \

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

/ / / / / / \ \ \ \ \ \

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

/ / / / / / / / / / / / /

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

\ \ \ \ \ \ / / / / / /

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

\ \ \ \ \ / / / / /

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

\ \ \ \ / / / /

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

\ \ \ / / /

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

\ \ / /

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

\ /

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

CROSSREFS

Cf. A001358, A001399, A113688.

For the sequence of isolated primes see A335916.

Related sequences:

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

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

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

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

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

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

Sequence in context: A335704 A292818 A173082 * A372336 A133395 A341424

Adjacent sequences: A113650 A113651 A113652 * A113654 A113655 A113656

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Jan 16 2006

EXTENSIONS

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.

Terms a(4) onwards corrected by R. J. Mathar, Jun 29 2020

STATUS

approved