A172979 -id:A172979 - cp's OEIS frontend

A078784 Primes on axis of Ulam square spiral (with rows ... / 7 8 9 / 6 1 2 / 5 4 3 / ... ) with origin at (1).

2, 11, 19, 23, 53, 61, 127, 139, 151, 163, 233, 281, 431, 541, 613, 743, 827, 977, 1009, 1279, 1621, 1871, 2003, 2281, 2377, 2731, 3109, 3221, 3511, 3571, 3631, 3691, 4001, 4129, 4523, 4591, 5077, 6361, 6521, 7789, 7877, 8419, 9851, 10151, 10973, 11503, 11719, 11827, 12377, 12601, 12713, 13399

Offset: 1

Donald S. McDonald, Jan 10 2003

nonn

Quadrants are numbered clockwise: 4=north, 1=east, 2=south, 3=west. The spiral numbers falling on axes (whether prime or not) are 4=north (2n+1)^2-n, 1=east (2n+1)^2+n+1, 2=south (2n)^2-(n-1), 3=west (2n)^2+n+1.
Primes to the left, right, above or below the 1 in the example in A054552.
This is the union of the primes in A168022, A168023, A168025 and A168027. - R. J. Mathar, Jul 11 2014

			For n=0, quadrant = 1, a(1) =  2, distance = 1;
for n=1, quadrant = 1, a(2) = 11, distance = 2;
for n=2, quadrant = 3, a(3) = 19, distance = 2.

Cf. A039823, A054552, A054556, A054567, A033951, A172979.

Select[ Sort@ Flatten@ Table[ 4n^2 + (2j - 3)n + 1, {j, 0, 3}, {n, 58}], PrimeQ] (* Robert G. Wilson v, Jul 10 2014 *)

Primes in A039823(n) = ceiling((n^2 + n + 2)/4). - Georg Fischer, Dec 04 2024

a(12) onward from Robert G. Wilson v, Jul 10 2014

A270269 Prime numbers with locations of right angle turns in the Ulam square spiral that are vertices of isosceles right triangles.

Original entry on oeis.org

3, 5, 7, 31, 37, 43, 8011, 8101, 8191, 920641, 921601, 922561, 3894703, 3896677, 3898651, 5902471, 5904901, 5907331, 7450171, 7452901, 7455631, 7482961, 7485697, 7488433, 36066031, 36072037, 36078043, 37155121, 37161217, 37167313, 39759331, 39765637, 39771943

Offset: 1

Michel Lagneau, Mar 14 2016

nonn

See the illustration for more information.
Subsequence of A172979. This sequence is probably infinite.
An interesting property: the sequence of the differences between prime numbers that are vertices for each triangle is the sequence {2, 6, 90, 960, 1974, 2430, 2730, 2736, 6006, 6096, 6306, ...} = A087277: numbers n such that the three second-degree cyclotomic polynomials x^2 + 1, x^2 - x + 1 and x^2 + x + 1 are simultaneously prime.
For example:
2 = 5 - 3 = 7 - 5;
6 = 37 - 31 = 43 - 37;
90 = 8101 - 8011 = 8191 - 8101.
Consequence: a(3n) + A087277(n) is a square. The corresponding sequence of the squares is {3^2, 7^2, 91^2, 961^2, 1975^2, 2431^2, 2731^2, 2737^2, 6007^2, ...}.
Examples:
a(3) + A087277(1) = 7 + 2 = 3^2;
a(6) + A087277(2) = 43 + 6 = 7^2;
a(9) + A087277(3) = 8191 + 90 = 91^2.

Michel Lagneau, Illustration

Cf. A087277, A172979.

nn:=20000:T:=array(1..nn):a0:=1:kk:=0:
for p from 1 to nn do :
   a1:=a0+floor(p/2):a0:=a1:
    if  isprime(a1)
     then
     kk:=kk+1:T[kk]:=a1:
     else
    fi:
  od:
   for n from 1 to kk-2 do:
    d1:=T[n+2]-T[n+1]:d2:=T[n+1]-T[n]:
     if d1=d2
     then
      printf("%d %d %d \n", T[n], T[n+1], T[n+2]):
      else
     fi:
   od:

A331997 Semiprimes m = p*q where m, p and q are in A033638 (locations of right angle turns in Ulam square spiral).

Original entry on oeis.org

10, 21, 26, 65, 91, 111, 785, 842, 1333, 4097, 21171, 28562, 50851, 100807, 194923, 970226, 1000001, 37021141, 65618101, 81144065, 151782401, 151819363, 174134417, 577921601, 688773781, 796622401, 796678851, 1276025563, 2090501285, 2176782337, 3662246773, 6059299123, 6879790081

Offset: 1

Michel Lagneau, Feb 04 2020

nonn

The sequence is probably infinite.
A geometric property of the sequence: consider the first diagonal with numbers of the form f(k) = k^2 + k + 1 in the Ulam spiral. The semiprimes and their prime factors belonging to the diagonal are given by the subsequence: 21, 91, 1333, 50851, 194923, 37021141, 65618101, 151819363, 688773781, 796622401, 1276025563, 3662246773, 6059299123, 6879790081, ... (see the illustration). This subsequence is the result of the following property: f(k)*f(k+1) = f((k+1)^2).
Examples:
21 = 3*7 = f(1)*f(2) = f(4);
91 = 7*13 = f(2)*f(3) = f(9);
1333 = 31*43 = f(5)*f(6) = f(36);
................................
This subsequence is probably infinite.

			111 is in the sequence because 111 = 3*37, and the numbers 3, 37 and 111 are in A033638.

Michel Lagneau, Ulam Spiral

Cf. A001358, A033638, A172979.

with(numtheory):nn:=10^5:T1:=1:
lst:={1}:lst1:={}:
  for n from 2 to nn do:
   T2:= T1 + floor(n/2):lst:=lst union {T2}:T1:=T2:
  od:
      for j from 2 to nn do:
        x:=lst[j]:d:=factorset(x):n0:=nops(d):
          if n0=2 and bigomega(x)=2
           and {d[1],d[2]} intersect lst = {d[1],d[2]}
           then
           lst1:=lst1 union {lst[j]}
           else
           fi:
          od:
          sort(lst1);

lista(nn) = {my(vn = vector(nn, k, k^2\4 + 1)); for (i=1, #vn, if (bigomega(vn[i]) == 2, my(f=factor(vn[i])); my(p=f[1,1], q = f[2,1]); if (vecsearch(vn, p) && vecsearch(vn, q), print1(vn[i], ", "));););} \\ Michel Marcus, Feb 04 2020

More terms from Michel Marcus, Jan 13 2025

cp's OEIS Frontend

A078784 Primes on axis of Ulam square spiral (with rows ... / 7 8 9 / 6 1 2 / 5 4 3 / ... ) with origin at (1).

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A270269 Prime numbers with locations of right angle turns in the Ulam square spiral that are vertices of isosceles right triangles.

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A331997 Semiprimes m = p*q where m, p and q are in A033638 (locations of right angle turns in Ulam square spiral).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

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Extensions