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

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..30.

Michel Lagneau, Ulam Spiral

Example

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

Maple

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);

Prog

(PARI) 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(vp, p) && vecsearch(vp, q), print1(vn[i], ", ")); ); ); } \\ Michel Marcus, Feb 04 2020

Crossrefs

Cf. A001358, A033638, A172979.

Sequence in context: A367576 A353402 A353432 * A363222 A242287 A165403

Adjacent sequences: A331994 A331995 A331996 * A331998 A331999 A332000

Keyword

nonn

Author

Michel Lagneau, Feb 04 2020

Status

approved