A113688 -id:A113688 - cp's OEIS frontend

A114412 Records in semiprime gaps ordered by merit.

2, 3, 4, 6, 11, 19, 24, 28, 30, 32, 38, 47, 54, 70, 74, 107, 110, 112, 120, 126, 146

Offset: 1

Jonathan Vos Post, Nov 25 2005

There is an associated index list n = 1, 2, 4, 6, 34, 422, 1765, 4585, 8112, 8650, 8861, 75150, ... and an associated semiprime list A001358(n) = 4, 6, 10, 15, 1418, 6559, 17965, 32777, 35103, 35981, 340894, ... - R. J. Mathar, Mar 15 2009

			Records defined in terms of A065516 and A001358:
.
  n  A065516(n)  A065516(n)/log_10(A001358(n))
  =  ==========  ==============================
  1       2      2 / log_10(4)  = 3.32192809...
  2       3      3 / log_10(6)  = 3.85529162...
  3       1      1 / log_10(9)  = 1.04795163...
  4       4      4 / log_10(10) = 4.00000000
  5       1      1 / log_10(14) = 0.87250286...
  6       6      6 / log_10(15) = 5.10164492...
  7       1      1 / log_10(21) = 0.75630419...
  8       3      3 / log_10(22) = 2.23476557...
  9       1      1 / log_10(25) = 0.71533827...

Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A001358, A065516, A111870, A111871, A113688-A113693, A114403-A114411, A114412-A114422.

sp = 4; m0 = 0;  l = {}; lim = 1000000;
For[i = 5, i <= lim, i++, If[PrimeOmega[i] == 2, m = (i - sp)/Log[sp]; If[m > m0, m0 = m; AppendTo[l, i - sp]]; sp = i] ]; l (* Robert Price, Oct 29 2018 *)

a(n) = records in A065516(n)/log_10(A001358(n)) = records in (A001358(n+1) - A001358(n))/log_10(A001358(n)).

Corrected and extended by Charles R Greathouse IV, Oct 05 2006

a(16)-a(21) from Donovan Johnson, Feb 17 2010

A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

Original entry on oeis.org

83, 101, 127, 137, 163, 199, 233, 311, 373, 443, 463, 491, 541, 587, 613, 631, 641, 659, 673, 683, 691, 733, 757, 797, 859, 881, 911, 919, 953, 971, 991, 1013, 1051, 1061, 1103, 1109, 1117, 1193, 1201, 1213, 1249, 1307, 1319, 1409, 1433, 1459, 1483, 1487

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Feb 17 2006

nonn

Isolated prime numbers have no adjacent primes in a lattice generated by writing consecutive integers starting from 1 in a spiral distribution. If n0 is the number of isolated primes and p the number of primes less than N, the ratio n0/p approaches 1 as N increases. If n1, n2, n3, n4 denote the number of primes with respectively 1, 2, 3, 4 adjacent primes in the lattice, the ratios n1/n0, n2/n1, n3/n2, n4/n3 approach 0 as N increases. The limits stand for any 2D lattice of integers generated by a priori criteria (i.e., not knowing distributions of primes) as Ulam's lattice.

			83 is an isolated prime as the adjacent numbers in lattice 50, 51, 81, 82, 84, 123, 124, 125 are not primes.
From _Michael De Vlieger_, Dec 22 2015: (Start)
Spiral including n <= 17^2 showing only primes, with the isolated primes in parentheses (redrawn by _Jon E. Schoenfield_, Aug 06 2017):
  257 .  .  .  .  . 251 .  .  .  .  .  .  .  .  . 241
   . 197 .  .  . 193 . 191 .  .  .  .  .  .  .  .  .
   .  .  .  .  .  .  .  . 139 .(137).  .  .  .  . 239
   .(199).(101).  .  . 97  .  .  .  .  .  .  . 181 .
   .  .  .  .  .  .  .  . 61  . 59  .  .  . 131 .  .
   .  .  . 103 . 37  .  .  .  .  . 31  . 89  . 179 .
  263 . 149 . 67  . 17  .  .  . 13  .  .  .  .  .  .
   .  .  .  .  .  .  .  5  .  3  . 29  .  .  .  .  .
   .  . 151 .  .  . 19  .  .  2 11  . 53  .(127).(233)
   .  .  . 107 . 41  .  7  .  .  .  .  .  .  .  .  .
   .  .  .  . 71  .  .  . 23  .  .  .  .  .  .  .  .
   .  .  . 109 . 43  .  .  . 47  .  .  .(83) . 173 .
  269 .  .  . 73  .  .  .  .  . 79  .  .  .  .  . 229
   .  .  .  .  . 113 .  .  .  .  .  .  .  .  .  .  .
  271 . 157 .  .  .  .  .(163).  .  . 167 .  .  . 227
   . 211 .  .  .  .  .  .  .  .  .  .  . 223 .  .  .
   .  .  .  . 277 .  .  . 281 . 283 .  .  .  .  .  .
(End)

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 22.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001107, A002939, A007742, A033951-A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848.

Cf. A113688 (isolated semiprimes in the semiprime spiral), A156859.

# A is Ulam's lattice
if (isprime(A[x,y])and(not(isprime(A[x+1,y]) or isprime(A[x-1,y])or isprime(A[x,y+1])or isprime(A[x,y-1])or isprime(A[x-1,y-1])or isprime(A[x+1,y+1])or isprime(A[x+1,y-1])or isprime(A[x-1,y+1])))) then print (A[x,y]) ; fi;

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@ # &]]; f[spiral@ 21 /. n_ /; CompositeQ@ n -> 0] (* Michael De Vlieger, Dec 22 2015, Version 10 *)

A113689 Number of semiprimes in clumps of size > 1 through n^2 in the semiprime spiral.

Original entry on oeis.org

0, 0, 2, 6, 9, 13, 17, 21, 23, 31, 37, 45, 54, 59, 72, 77, 83, 93, 104, 116, 125, 140, 150, 164, 180, 188, 203, 219, 236, 255, 272, 287, 301, 317, 334, 354, 378, 403, 419, 430, 450, 475, 498, 521, 542, 560, 588, 608, 626, 652, 677, 698

Offset: 1

Jonathan Vos Post, Nov 05 2005

Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. Analogous to Ulam coloring in the primes in the spiral and discovering unexpectedly many connected diagonals, we construct a semiprime spiral by coloring in all 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, A113689, gives an enumeration of the number of semiprimes in clumps of size > 1 through n^2, not looking past the square boundary. A113688 gives isolated semiprimes in the semiprime spiral, namely those semiprimes none of whose adjacent integers in the spiral are semiprimes.

			a(3) = 2 because there is one visible clump through 3^2 = 9, {4,6}, which two semiprimes are diagonally connected.
a(4) = 6 because there are 6 semiprimes in the 2 visible clumps through 4^2 = 16, {4, 6, 14, 15}, {9, 10}.
a(5) = 9 because there are 9 semiprimes in the 3 visible clumps through 5^2 = 25, {4, 6, 14, 15}, {9, 10, 25}, {21, 22}.
......................
... 17 16 15 14 13 ...
... 18  5  4  3 12 ...
... 19  6  1  2 11 ...
... 20  7  8  9 10 ...
... 21 22 23 24 25 ...
......................

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

M. Stein and S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74, 43-44, 1967.
M. Stein, S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, Amer. Math. Monthly 71, 516-520, 1964.
Eric Weisstein's World of Mathematics, Prime Spiral.
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826, A113688.

Corrected and extended by Alois P. Heinz, Jan 02 2011

A113693 Semiprimes in A054556.

Original entry on oeis.org

4, 15, 34, 249, 391, 565, 771, 886, 1915, 3814, 5149, 5739, 6046, 7354, 9169, 10765, 11611, 15814, 16321, 18429, 20665, 22426, 24259, 28141, 29499, 32311, 36769, 39106, 43161, 48291, 52786, 53709, 57481, 60394, 63379, 65409, 67471, 69565

Offset: 1

Jonathan Vos Post, Nov 05 2005

This sequence contains semiprimes from the center straight up the y-axis in the semiprime spiral of A113688-A113689. Semiprimes from the center straight down the y-axis in the semiprime spiral are A113691. Semiprimes from the center straight right along the x-axis in the semiprime spiral are A113690. Semiprimes from the center straight left along the x-axis in the semiprime spiral are A113692.

			a(27) = 4*97^2 - 9*97 + 6 = 36769 = 83 * 443.
a(28) = 4*100^2 - 9*100 + 6 = 39106 = 2 * 19553.
a(27) and a(28) are horizontally adjacent in the prime spiral, hence part of a clump and not isolated semiprimes as in A113688.
a(45) = 4*157^2 - 9*157 + 6 = 97189 = 17 * 5717 is the greatest member under 10^5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001358, A054556, A113688-A113699.

IsSemiprime:= func; [s: n in [2..150] | IsSemiprime(s) where s is 4*n^2 - 9*n + 6]; // Vincenzo Librandi, Sep 22 2012

Select[Table[4 n^2 - 9 n + 6, {n, 140}], PrimeOmega[#] == 2 &] (* Vincenzo Librandi, Sep 22 2012 *)

{a(n)} = Intersection of A001358 and A054556. Semiprimes of the form 4*k^2 - 9*k + 6.

A113690 Semiprimes in A054552.

Original entry on oeis.org

86, 298, 371, 1243, 1541, 2426, 2627, 3053, 4258, 5366, 5663, 6281, 6602, 6931, 7613, 8327, 9073, 9458, 10661, 13283, 14702, 15191, 16706, 18293, 18838, 23486, 25361, 26002, 26651, 27973, 28646, 34318, 35063, 36577, 38123, 41311, 43786, 44627

Offset: 1

Jonathan Vos Post, Nov 05 2005

This sequence, A113690, contains semiprimes from the center straight right along the x-axis in the semiprime spiral of A113688-A113689. Semiprimes from the center straight left along the x-axis in the semiprime spiral are A113692. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral. Semiprimes from the center straight down the y-axis in the semiprime spiral are A113691.

			a(10) = 4*37^2 - 3*37 + 1 = 5366 = 2 * 2683.
a(11) = 4*38^2 - 3*38 + 1 = 5663 = 7 * 809.
a(10) and a(11) are horizontally adjacent in the prime spiral, hence part of a clump and not isolated semiprimes as in A113688.
a(57) = 4*156^2 - 3*156 + 1 = 96877 = 11 * 8807 is the greatest member under 10^5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001358, A054552, A113688-A113699.

IsSemiprime:= func; [s: n in [1..120] | IsSemiprime(s) where s is 4*n^2 - 3*n + 1]; // Vincenzo Librandi, Sep 22 2012

Select[Table[4*n^2 - 3*n + 1, {n, 150}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

{a(n)} = Intersection of A001358 and A054552. Semiprimes of the form 4*k^2 - 3*k + 1.

Corrected a(6) by Vincenzo Librandi, Sep 22 2012

A113691 Semiprimes in A033951.

Original entry on oeis.org

46, 77, 218, 1073, 1351, 1502, 1661, 2186, 2998, 4193, 4727, 5006, 5293, 5891, 7183, 8603, 10558, 12266, 13631, 14581, 15563, 19811, 20953, 25202, 27806, 29843, 30538, 31241, 32671, 33398, 35627, 37153, 39502, 40301, 46118, 46981, 49618, 56051

Offset: 1

Jonathan Vos Post, Nov 05 2005

This sequence, A113691, contains semiprimes from the center straight down the y-axis in the semiprime spiral of A113688-A113689. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral. A113690 contains semiprimes from the center straight right along the x-axis in the semiprime spiral. Semiprimes from the center straight left along the x-axis in the semiprime spiral are A113692.

			a(5) = 4*18^2 + 3*18 + 1 = 1351 = 7 * 193.
a(6) = 4*19^2 + 3*19 + 1 = 1502 = 2 * 751.
a(7) = 4*20^2 + 3*20 + 1 = 1661 = 11 * 151.
a(5), a(6) and a(7) are vertically adjacent in the semiprime spiral, hence part of a clump and not isolated semiprimes as in A113688. a(11), a(12) and a(13) are another such vertical string of 3 adjacent semiprimes and so is a(26), a(27) and a(28).
a(52) = 4*152^2 + 3*152 + 1 = 92873 = 11 * 8443 is the greatest member under 10^5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001358, A033951, A113688-A113699.

IsSemiprime:= func; [s: n in [1..120] | IsSemiprime(s) where s is 4*n^2 + 3*n + 1]; // Vincenzo Librandi, Sep 22 2012

Select[Table[4*n^2 + 3*n + 1, {n, 200}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

{a(n)} = Intersection of A001358 and A033951. Semiprimes of the form 4*k^2 + 3*k + 1.

A113692 Semiprimes in A054567.

Original entry on oeis.org

6, 69, 106, 265, 334, 411, 589, 799, 1041, 1174, 1315, 1959, 2329, 3394, 4659, 5221, 5815, 7099, 8146, 8511, 10869, 16449, 21979, 23181, 23794, 25681, 26326, 31774, 33949, 35439, 36961, 38515, 40101, 43369, 45051, 48511, 50289, 52099, 54874

Offset: 1

Jonathan Vos Post, Nov 05 2005

This sequence, A113692, contains semiprimes from the center straight left along the x-axis in the semiprime spiral of A113688-A113689. A113690 contains semiprimes from the center straight right along the x-axis in the semiprime spiral. A113691 contains semiprimes from the center straight down the y-axis in the semiprime spiral. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral.

			a(4) = 4*9^2 - 7*9 + 4 = 265 = 5 * 53.
a(5) = 4*10^2 - 7*10 + 4 = 334 = 2 * 167.
a(6) = 4*11^2 - 7*11 + 4 = 411 = 3 * 137.
a(4), a(5) and a(6) are horizontally adjacent in the semiprime spiral, hence part of a clump and not isolated semiprimes as in A113688. a(9), a(10) and a(11) are another such horizontal string of 3 adjacent semiprimes.
a(46) = 4*151^2 - 7*151 + 4 = 90151 = 17 * 5303 is the greatest member under 10^5 (it is coincidence that this integer ends, base 10, with the same 151 that is the index of the quadratic).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001358, A054567, A113688-A113699.

IsSemiprime:= func; [s: n in [2..120] | IsSemiprime(s) where s is 4*n^2 - 7*n + 4]; // Vincenzo Librandi, Sep 22 2012

Select[Table[4*n^2 - 7*n + 4, {n, 200}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

{a(n)} = Intersection of A001358 and A054567. Semiprimes of the form 4*k^2 - 7*k + 4.

A113653 Isolated semiprimes in the hexagonal spiral.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jan 16 2006

nonn

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

			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

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

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.

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).

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

A335704 Erroneous version of A113653.

Original entry on oeis.org

6, 51, 55, 69, 82, 183, 194, 249, 259, 287, 309, 314, 319

Offset: 1

dead

This is the erroneous version of A113653 that was submitted to the OEIS by Jonathan Vos Post on Jan 16 2006. Because 44 was omitted from the spiral, not only are the terms here incorrect, but a large number of other sequences will also need to be corrected. For this reason the whole of the original submission has been preserved here with a different A-number. - N. J. A. Sloane, Jun 27 2020

Isolated semiprimes in the hexagonal spiral, embedded in the triangular lattice, are the analogy to A113688 "Isolated semiprimes in the [square] spiral," as well as analogous in another way to the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld]. A113519 Semiprimes in first spoke of a hexagonal spiral (A056105). A113524 Semiprimes in second spoke of a hexagonal spiral (A056106). A113525 Semiprimes in third spoke of a hexagonal spiral (A056107). A113527 Semiprimes in fourth spoke of a hexagonal spiral (A056108). A113528 Semiprimes in fifth spoke of a hexagonal spiral (A056109). A113530 Semiprimes in sixth spoke of a hexagonal spiral (A003215). This is embedded in the hexagonal spiral of A003215 and A001399, which is centered on zero; of course such a spiral can be constructed beginning with any integer. Centering on zero gives the interesting partition and multigraph equalities of A001399.

			Copy this as proportionally spaced text, make semiprimes bold, draw boundaries around clumps of adjacent semiprimes. For example, there is a triangular clump of three semiprimes: {4, 14, 15}; a linear clump of three semiprimes {49, 77, 111}; a linear clump of two semiprimes {247, 305}; an irregular clump of seven {115, 155, 201, 202, 203, 253, 254}; a clump of eighteen whose least element is 33 and greatest is 206; and a long branching clump of sixteen whose least element is 9 and greatest is 129.
.................209.208.207.206.205.204.203.202.201
................210.162.161.160.159.158.157.156.155.200
..............211.163.121.120.119.118.117.116.115.154.199
............212.164.122.86..85..84..83..82..81.114.153.198
..........213.165.123.87..57..56..55..54..53..80.113.152.197
........214.166.124.88..58..33..32..31..30..52..79.112.151.196
......215.167.125.89..59..34..16..15..14..29..51..78.111.150.195
....216.168.126.90..60..35..17..5...4...13..28..50..77.110.149.194
..217.169.127.91..61..36..18..6...0...3...12..27..49..76.109.148.193
218.170.128.92..62..37..19..7...1...2...11..26..48..75.108.147.192.243
..219.171.129.93..63..38..20..8...9...10..25..47..74.107.146.191.242
....220.172.130.94..64..39..21..22..23..24..46..73.106.145.190.241
......221.173.131.95..65..40..41..42..43..45..72.105.144.189.240
........222.174.132.96..66..67..68..69..70..71.104.143.188.239
..........223.175.133.97..98..99.100.101.102.103.142.187.238
............224.176.134.135.136.137.138.139.140.141.186.237
..............225.177.178.179.180.181.182.183.184.185.236
................226.227.228.229.230.231.232.233.234.235

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

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A003215, A056105-A056109, A113688, A113519, A113524, A113525, A113528, A113527, A113530, A113688.

{a(n)} = {integers in A001358 which are not adjacent in any of six directions to any other integers in A001358 when arranged as the hexagonal spiral of A003215}.

cp's OEIS Frontend

A114412 Records in semiprime gaps ordered by merit.

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A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

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A113689 Number of semiprimes in clumps of size > 1 through n^2 in the semiprime spiral.

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A113693 Semiprimes in A054556.

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Magma

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A113690 Semiprimes in A054552.

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Magma

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A113691 Semiprimes in A033951.

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A113692 Semiprimes in A054567.

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