cp's OEIS Frontend

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

%I A331997 #17 Jan 13 2025 12:31:55
%S A331997 10,21,26,65,91,111,785,842,1333,4097,21171,28562,50851,100807,194923,
%T A331997 970226,1000001,37021141,65618101,81144065,151782401,151819363,
%U A331997 174134417,577921601,688773781,796622401,796678851,1276025563,2090501285,2176782337,3662246773,6059299123,6879790081
%N A331997 Semiprimes m = p*q where m, p and q are in A033638 (locations of right angle turns in Ulam square spiral).
%C A331997 The sequence is probably infinite.
%C A331997 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).
%C A331997 Examples:
%C A331997 21 = 3*7 = f(1)*f(2) = f(4);
%C A331997 91 = 7*13 = f(2)*f(3) = f(9);
%C A331997 1333 = 31*43 = f(5)*f(6) = f(36);
%C A331997 ................................
%C A331997 This subsequence is probably infinite.
%H A331997 Michel Lagneau, <a href="/A331997/a331997.pdf">Ulam Spiral</a>
%e A331997 111 is in the sequence because 111 = 3*37, and the numbers 3, 37 and 111 are in A033638.
%p A331997 with(numtheory):nn:=10^5:T1:=1:
%p A331997 lst:={1}:lst1:={}:
%p A331997   for n from 2 to nn do:
%p A331997    T2:= T1 + floor(n/2):lst:=lst union {T2}:T1:=T2:
%p A331997   od:
%p A331997       for j from 2 to nn do:
%p A331997         x:=lst[j]:d:=factorset(x):n0:=nops(d):
%p A331997           if n0=2 and bigomega(x)=2
%p A331997            and {d[1],d[2]} intersect lst = {d[1],d[2]}
%p A331997            then
%p A331997            lst1:=lst1 union {lst[j]}
%p A331997            else
%p A331997            fi:
%p A331997           od:
%p A331997           sort(lst1);
%o A331997 (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(vn, p) && vecsearch(vn, q), print1(vn[i], ", "));););} \\ _Michel Marcus_, Feb 04 2020
%Y A331997 Cf. A001358, A033638, A172979.
%K A331997 nonn
%O A331997 1,1
%A A331997 _Michel Lagneau_, Feb 04 2020
%E A331997 More terms from _Michel Marcus_, Jan 13 2025