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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 04 2020
STATUS
approved