A033951 -id:A033951 - cp's OEIS frontend

A054925 a(n) = ceiling(n*(n-1)/4).

0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, 716, 743, 770, 798

Offset: 0

N. J. A. Sloane, May 24 2000

Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
Form the clockwise spiral starting 0,1,2,....; then A054925(n+1) interleaves 2 horizontal (A033951, A033991) and 2 vertical (A007742, A054552) branches. A bisection is A014848. - Paul Barry, Oct 08 2007
Consider the standard 4-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 2 along the positive y-axis, 3 along the positive z-axis, 4 along the positive t-axis, and then back round to the x-axis. This sequence gives the floor of the Euclidean distance to the origin after n steps. - Jon Perry, Apr 16 2013
Jon Perry's JavaScript code is explained by A238604. - Michael Somos, Mar 01 2014
Ceiling of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Ceiling of one-half of each triangular number. - Harvey P. Dale, Oct 03 2016
For n > 2, also the edge cover number of the (n-1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
Conjecture: For n>11, there always exists a prime number p such that a(n)Raul Prisacariu, Sep 01 2024
For n = 1 up to at least n = 13, also the lower matching number of the triangular honeycomb bishop graph. - Eric W. Weisstein, Dec 13 2024
Conjecturally, apart from the first term, the sequence terms are the exponents in the expansion of Sum_{n >= 0} q^(3*n) * (Product_{k >= 2*n+1} 1 - q^k) = 1 - q - q^2 + q^3 + q^5 - q^8 - q^11 + + - - .... Cf. A039825. - Peter Bala, Apr 13 2025

			a(6) = 8; ceiling(6*(6-1)/4) = ceiling(30/4) = 8.
G.f. = x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 18*x^9 + 23*x^10 + ...

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Craig Knecht, Binary fill patterns in a triangle refilled with natural numbers.
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A007742, A033951, A033991, A039825, A054552, A054924, A054925 + A011848 = C(n, 2).

Cf. A213172, A238604.

p=new Array(0,0,0,0);
for (a=0;a<100;a++) {
p[a%4]+=a;
document.write(Math.floor(Math.sqrt(p[0]*p[0]+p[1]*p[1]+p[2]*p[2]+p[3]*p[3]))+", ");
} /* Jon Perry, Apr 16 2013 */

[ Ceiling(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

I:=[0,0,1,2,3]; [n le 5 select I[n] else 3*Self(n-1)-4*Self(n-2)+4*Self(n-3)-3*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

seq(ceil(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Ceiling[(n^2 - n)/4], {n, 0, 20}] (* Wesley Ivan Hurt, Nov 01 2013 *)
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 0, 1, 2, 3}, 60] (* Vincenzo Librandi, Jul 14 2015 *)
Join[{0}, Ceiling[#/2] &/ @ Accumulate[Range[0, 60]]] (* Harvey P. Dale, Oct 03 2016 *)
Ceiling[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Dec 13 2024 *)
Table[Ceiling[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
Table[(1 + (n - 1) n - Cos[n Pi/2] - Sin[n Pi/2])/4, {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
CoefficientList[Series[x^2 (-1 + x - x^2)/((-1 + x)^3 (1 + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 13 2024 *)

{a(n) = ceil( n * (n-1)/4)}; /* Michael Somos, Feb 11 2004 */

[ceil(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - Michael Somos, Sep 02 2006
From Michael Somos, Feb 11 2004: (Start)
G.f.: x^2 * (x^2 - x + 1) / ((1 - x)^3 * (1 + x^2)) = x^2 * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)).
a(1-n) = a(n).
A011848(n) = a(-n). (End)
From Michael Somos, Mar 01 2014: (Start)
a(n + 4) = a(n) + 2*n + 3.
a(n+1) = floor( sqrt( A238604(n))). (End)
a(n) = A011848(n) + A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
Sum_{n>=2} 1/a(n) = 4 - Pi + 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(1/sqrt(2)+cosh(sqrt(7)*Pi/4))). - Amiram Eldar, Dec 23 2024

A033990 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative y-axis.

Original entry on oeis.org

0, 1, 1, 8, 3, 7, 6, 2, 1, 5, 1, 1, 6, 2, 2, 1, 3, 4, 0, 4, 5, 3, 6, 7, 0, 8, 9, 1, 4, 6, 1, 2, 7, 1, 1, 4, 4, 8, 1, 7, 4, 7, 2, 0, 8, 8, 2, 4, 4, 1, 2, 8, 4, 6, 3, 2, 7, 3, 3, 7, 3, 2, 4, 1, 2, 3, 4, 7, 5, 6, 5, 2, 0, 1, 5, 8, 9, 8, 6, 4, 1, 7, 6, 1, 7, 8, 7, 7, 5, 1, 8, 4, 7, 6, 9, 2, 2, 3, 9, 0, 1, 0, 1, 6, 8

Offset: 0

N. J. A. Sloane

Consider array of digits 0_(1)23456789(1)0111213141516171(8)1920212223...; in this array add to n-th pointer 8*n+1 to get next pointer. E.g., n=1 so n+(8*1+1)=10 -> n=10 so n+(8*2+1)=27 -> n=27 so ... etc. - comment from Patrick De Geest.

			The spiral begins
                 2---3---2---4---2---5---2
                 |                       |
                 2   1---3---1---4---1   6
                 |   |               |   |
                 2   2   4---5---6   5   2
                 |   |   |       |   |   |
                 1   1   3   0   7   1   7
                 |   |   |   |   |   |   |
                 2   1   2---1   8   6   2
                 |   |           |   |   |
                 0   1---0---1---9   1   8
                 |                   |   |
                 2---9---1---8---1---7   2
                                         |
                             3---0---3---9
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading downwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033988, A033989. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2 - 3n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2-3*n-1) for n > 0. - Andrew Woods, May 20 2012

More terms from Patrick De Geest, Oct 15 1999

Edited by Charles R Greathouse IV, Nov 01 2009

A033953 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

Original entry on oeis.org

0, 7, 1, 7, 4, 2, 8, 1, 1, 3, 1, 2, 0, 2, 3, 1, 3, 4, 6, 5, 5, 5, 7, 7, 8, 8, 9, 6, 8, 1, 1, 1, 2, 3, 1, 8, 0, 6, 1, 7, 0, 9, 2, 8, 4, 3, 2, 1, 1, 7, 2, 6, 2, 1, 3, 3, 5, 5, 3, 2, 2, 0, 4, 3, 2, 5, 4, 6, 5, 0, 5, 1, 1, 6, 5, 8, 1, 2, 6, 7, 3, 8, 7, 8, 9, 5, 7, 1, 8, 2, 8, 6, 1, 9, 9, 3, 6, 7, 9, 0, 1, 4, 6, 1, 0

Offset: 0

N. J. A. Sloane

			  2---3---2---4---2---5---2
  |                       |
  2   1---3---1---4---1   6
  |   |               |   |
  2   2   4---5---6   5   2
  |   |   |       |   |   |
  1   1   3   0   7   1   7
  |   |   |   |   |   |   |
  2   1   2---1   8   6   2
  |   |           |   |   |
  0   1---0---1---9   1   8
  |                   |   |
  2---9---1---8---1---7   2
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading rightwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Sequences based on the same spiral: A033988, A033989, A033990. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]];
a[n_] := If[n==0, 0, A033307[[4n^2 + 3n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2 + 3*n - 1) for n > 0. - Andrew Woods, May 20 2012

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A113688 Isolated semiprimes in the semiprime square spiral.

Original entry on oeis.org

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

A014848 a(n) = n^2 - floor( n/2 ).

Original entry on oeis.org

0, 1, 3, 8, 14, 23, 33, 46, 60, 77, 95, 116, 138, 163, 189, 218, 248, 281, 315, 352, 390, 431, 473, 518, 564, 613, 663, 716, 770, 827, 885, 946, 1008, 1073, 1139, 1208, 1278, 1351, 1425, 1502, 1580, 1661, 1743, 1828, 1914, 2003, 2093, 2186, 2280, 2377, 2475

Offset: 0

N. J. A. Sloane

Quasipolynomial of order 2. - Charles R Greathouse IV, Jan 19 2012

The binomial transform is 0, 1, 5, 20,... which is A084850 with offset 1. - R. J. Mathar, Nov 26 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A033951, A033991, A042963 (first differences), A084850.

[n^2-Floor(n/2) : n in [0..50]]; // Wesley Ivan Hurt, Nov 14 2014

A014848:=n->n^2 - floor(n/2); seq(A014848(n), n=0..50); # Wesley Ivan Hurt, Oct 11 2013

Table[n^2-Floor[n/2],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
LinearRecurrence[{2,0,-2,1},{0,1,3,8},60] (* Harvey P. Dale, Jun 13 2022 *)

PARI
```
a(n)=n^2-n\2
```

def A014848(n): return n**2-(n>>1) # Chai Wah Wu, Jan 18 2023

[n^2 - (n//2) for n in range(71)] # G. C. Greubel, Mar 14 2024

a(2*n) = A033991(n).
a(2*n+1) = A033951(n).
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1-x^2)).
a(n) = (2*n*(2*n-1) + 1 - (-1)^n)/4. - Bruno Berselli, Feb 17 2011
a(n) = round(n/(exp(1/n) - 1)), n > 0. - Richard R. Forberg, Nov 14 2014
E.g.f.: (1/4)*((1 + 2*x + 4*x^2)*exp(x) - exp(-x)). - G. C. Greubel, Mar 14 2024

A174114 Even central polygonal numbers (A193868) divided by 2.

Original entry on oeis.org

1, 2, 8, 11, 23, 28, 46, 53, 77, 86, 116, 127, 163, 176, 218, 233, 281, 298, 352, 371, 431, 452, 518, 541, 613, 638, 716, 743, 827, 856, 946, 977, 1073, 1106, 1208, 1243, 1351, 1388, 1502, 1541, 1661, 1702, 1828, 1871, 2003, 2048, 2186, 2233, 2377, 2426, 2576

Offset: 1

Reinhard Zumkeller, Mar 08 2010

Central terms of A170950, seen as a triangle of rows with an odd number of terms.
Equivalently, numbers of the form m*(4*m+3)+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Jan 05 2016
Conjecure: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n} 1 - q^k) = q + q^2 + q^8 + q^11 + q^23 + q^28 + .... Cf. A266883. - Peter Bala, May 10 2025

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A002061, A002522, A010696, A170950, A193868, A266883.

Cf. A033951: numbers of the form m*(4*m+3)+1 for nonnegative m.

Select[Table[(n (n + 1)/2 + 1)/2, {n, 600}], IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)
(Select[PolygonalNumber@ Range@ 100, OddQ] + 1 )/2 (* Version 10.4, or *)
Rest@ CoefficientList[Series[-x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)

a(n)=(2*n-1)*(2*n-1-(-1)^n)\4+1 \\ Charles R Greathouse IV, Jun 11 2015

a(n+3) - a(n+2) - a(n+1) + a(n) = A010696(n+1).
a(n) = A170950(A002061(n)).
a(n) = A193868(n)/2. - Omar E. Pol, Aug 16 2011
G.f.: -x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 18 2011
E.g.f.: ((2 + x + 2*x^2)*cosh(x) + (1 - x + 2*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Nov 16 2024
Sum_{n>=1} 1/a(n) = 4*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 12 2025

New name from Omar E. Pol, Aug 16 2011

A054570 Prime number spiral (clockwise, West spoke).

Original entry on oeis.org

2, 19, 83, 199, 389, 641, 967, 1361, 1823, 2377, 3001, 3709, 4517, 5419, 6353, 7477, 8623, 9791, 11159, 12577, 14083, 15667, 17417, 19273, 21149, 23063, 25229, 27431, 29683, 32183, 34543, 37171, 39877, 42641, 45599, 48673, 51719, 54973, 58171

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, West.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A033951, A054551, A054553, A054555, A054564, A054566, A054570, A053999.

Table[ Prime[4n^2 + 3n + 1], {n, 0, 40} ]

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is West, clockwise.

a(n) = A000040(A033951(n)). - R. J. Mathar, Aug 29 2018

Edited by Frank Ellermann, Feb 24 2002

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

A168027 Noncomposite numbers in the southern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 23, 163, 281, 431, 613, 827, 2003, 2377, 3221, 3691, 6521, 7877, 10151, 10973, 11827, 12713, 17623, 18701, 23333, 24571, 25841, 27143, 28477, 38711, 43577, 45263, 48731, 50513, 65921, 72227, 81083, 85703, 95327, 97813, 102881, 124433

Offset: 0

Alonso del Arte, Nov 16 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alonso del Arte, Ulam spiral (2009). [Note that "East" and "West" in this video match the cover of Scientific American, but "North" and "South" are switched.]
BackIssues.com, Scientific American March 1964 back issue
MathWorld, Prime Spiral
Scientific American, March 1964 cover
Wikipedia, Ulam Spiral

Cf. A033951, all numbers of the form 4n^2 + 3n + 1. Noncomposites of the eastern ray are in A168022. Primes of the northeastern ray are in A073337. Noncomposites of the northern ray are in A168023. Primes of the northwestern ray are in A121326. Noncomposites of the western ray are in A168025. Noncomposites of the southwestern ray are in A168026. There are no primes on the southeastern ray, which, being A016754, are the odd squares, and thus none of them are prime.

Select[Table[4 n^2 + 3 n + 1, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]

Positive numbers of the form 4n^2 + 3n + 1 with no more than two divisors.

A265409 a(n) = index to the nearest inner neighbor in Ulam-style square-spirals using zero-based indexing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 30, 30, 31, 32, 33, 34, 35, 36, 36, 36, 37, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 47, 48, 49, 49, 49, 50

Offset: 1

Antti Karttunen, Dec 13 2015

nonn

Each n occurs A265411(n+1) times.

Useful when defining recurrences like A078510 and A265408.

			We arrange natural numbers as a counterclockwise spiral into the square grid in the following manner (here A stands for 10, B for 11). The first square corresponds with n (where the initial term 0 is at the center), and the second square with the value of a(n). This sequence doesn't specify a(0), thus it is shown as an asterisk (*):
                    44322
            432B    40002B
            501A    50*01A
            6789    600119
                    667899
-
For each n > 0, we look for the nearest horizontally or vertically adjacent neighbor of n towards the center that is not n-1, which will then be value of a(n) [e.g., it is 0 for 3, 5 and 7, while it is 1 for 8, 9 and A (10) and 2 for B (11)] unless n is in the corner (one of the terms of A002620), in which case the value is the nearest diagonally adjacent neighbor towards the center, e.g. 0 for 2, 4 and 6, while it is 1 for 9).
See also the illustration at A078510.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One less than A265410(n+1).
Cf. A002620, A240025, A265359.
Cf. also A033951, A063826, A078510, A265408.

If n <= 7, a(n) = 0 for n >= 8: if either A240025(n) or A240025(n-1) is not zero [when n or n-1 is in A002620], then a(n) = a(n-1), otherwise, a(n) = 1 + a(n-1).
If n <= 7, a(n) = 0, for n >= 8, a(n) = a(n-1) + (1-A240025(n))*(1-A240025(n-1)). [The same formula in a more compact form.]
a(n) = A265410(n+1) - 1.
Other identities. For all n >= 0:
a(n) = n - A265359(n).

cp's OEIS Frontend

A054925 a(n) = ceiling(n*(n-1)/4).

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A033990 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative y-axis.

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A033953 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

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A113688 Isolated semiprimes in the semiprime square spiral.

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A014848 a(n) = n^2 - floor( n/2 ).

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A174114 Even central polygonal numbers (A193868) divided by 2.

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A054570 Prime number spiral (clockwise, West spoke).