A113689 -id:A113689 - cp's OEIS frontend

A113688 Isolated semiprimes in the semiprime square spiral.

65, 74, 249, 295, 309, 355, 422, 511, 545, 667, 669, 758, 926, 943, 979, 998, 1099, 1167, 1186, 1322, 1457, 1469, 1561, 1585, 1658, 1711, 1774, 1779, 1835, 1891, 1959, 1961, 1963, 2021, 2038, 2066, 2155, 2186, 2191, 2206, 2271, 2329, 2342

Offset: 1

Jonathan Vos Post, Nov 05 2005

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.

The squares of twin primes occupy adjacent points along the southeast diagonal, so none are isolated. Thus the only isolated semiprimes in the spiral that are squares are the squares of "isolated primes" (A007510). The first square in this sequence is a(1473) = 66049 = 257^2. - Jon E. Schoenfield, Aug 12 2018

			Spiral example:
.
  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
.
From _Michael De Vlieger_, Dec 22 2015: (Start)
Spiral including n <= 121 showing only semiprimes; the isolated semiprimes appear in parentheses:
.
    .---.---.---.---.---.--95--94--93---.--91
    |                                       |
    . (65)--.---.--62---.---.---.--58--57   .
    |   |                               |   |
    .   .   .---.--35--34--33---.---.   .   .
    |   |   |                       |   |   |
    .   .  38   .---.--15--14---.   .  55   .
    |   |   |   |               |   |   |   |
    .   .  39   .   .---4---.   .   .   .  87
    |   |   |   |   |       |   |   |   |   |
  106  69   .   .   6   .---.   .   .   .  86
    |   |   |   |   |           |   |   |   |
    .   .   .   .   .---.---9--10   .   .  85
    |   |   |   |                   |   |   |
    .   .   .  21--22---.---.--25--26  51   .
    |   |   |                           |   |
    .   .   .---.---.--46---.---.--49---.   .
    |   |                                   |
    .   .-(74)--.---.--77---.---.---.---.--82
    |
  111---.---.---.-115---.---.-118-119---.-121
.
(End)

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

Michael De Vlieger, Table of n, a(n) for n = 1..2000
Alois P. Heinz, Plot of semiprime spiral, containing all semiprimes <= 10000. Isolated semiprimes are colored red.
M. Stein and S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74, 43-44, 1967.
M. Stein and 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.

Cf. A115258 (isolated primes in Ulam's lattice).

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 *)

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.

cp's OEIS Frontend

A113688 Isolated semiprimes in the semiprime square spiral.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A113693 Semiprimes in A054556.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A113690 Semiprimes in A054552.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A113691 Semiprimes in A033951.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A113692 Semiprimes in A054567.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula