A379162 -id:A379162 - cp's OEIS frontend

A379532 Ulam numbers that are products of exactly four distinct primes (or tetraprimes).

390, 546, 690, 798, 1155, 1230, 1770, 2010, 2090, 2418, 2618, 2814, 3090, 3290, 3390, 3930, 4326, 4370, 4470, 4578, 4602, 4641, 6110, 6870, 7170, 7490, 7735, 7930, 8294, 9834, 10110, 10545, 10738, 11102, 11346, 11390, 11454, 11622, 11715, 11886, 12270, 12441, 12470, 12570

Offset: 1

Massimo Kofler, Dec 24 2024

nonn

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

			390 is a term because 390=2*3*5*13 is the product of 4 distinct primes and 390 is an Ulam number.
546 is a term because 546=2*3*7*13 is the product of 4 distinct primes and 546 is an Ulam number.
1155 is a term because 1155=3*5*7*11 is the product of 4 distinct primes and 1155 is an Ulam number.

Robert Israel, Table of n, a(n) for n = 1..6826

Intersection of A002858 and A046386.

Cf. A068820, A378795, A379162.

N:= 20000: # for terms <= N
U:= [1,2]: V:= Vector(N): V[3]:= 1: R:= NULL: count:= 0:
for i from 3 do
   for k from U[-1]+1 to N do
     if V[k] = 1 then
       J:= select(`<=`,U +~ k, N);
       V[J]:= V[J] +~ 1;
       U:= [op(U),k];
       F:= ifactors(k)[2]:
       if F[..,2] = [1,1,1,1] then R:= R,k; count:= count+1;  fi;
       break
     fi
   od;
   if k > N then break fi;
od:
R; # Robert Israel, Dec 25 2024

seq[numUlams_] := Module[{ulams = {1, 2}}, Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {numUlams}]; Select[ulams, FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &]]; seq[1200] (* Amiram Eldar, Dec 24 2024, after Jean-François Alcover at A002858 *)

A381189 Ulam numbers that are squarefree semiprimes.

Original entry on oeis.org

6, 26, 38, 57, 62, 69, 77, 82, 87, 106, 145, 155, 177, 206, 209, 219, 221, 253, 309, 319, 339, 341, 358, 382, 451, 485, 497, 502, 566, 685, 695, 734, 781, 849, 866, 893, 905, 949, 1018, 1037, 1079, 1081, 1101, 1157, 1167, 1169, 1186, 1191, 1257, 1313, 1355, 1387, 1389

Offset: 1

Massimo Kofler, Feb 16 2025

nonn

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

Number such as 4, 2809, 3481, 6889, etc., are in A378795 but not in this or in a longer sequence.

			4 is not a term of this sequence because 4=2^2 is the product of 2 not distinct primes even if 4 is an Ulam number.
6 is a term because 6=2*3 is the product of 2 distinct primes and 6 is an Ulam number.
57 is a term because 57=3*19 is the product of 2 distinct primes and 57 is an Ulam number.
2809 is not a term of a longer sequence because 2809=53^2 is the product of 2 not distinct primes even if 2809 is an Ulam number.

Intersection of A006881 and A002858.

Cf. A068820, A378795, A379162, A379532.

seq[numUlams_] := Module[{ulams = {1, 2}}, Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {numUlams}]; Select[ulams, FactorInteger[#][[;; , 2]] == {1, 1} &]]; seq[160] (* Amiram Eldar, Feb 16 2025, after Jean-François Alcover at A002858 *)

cp's OEIS Frontend

A379532 Ulam numbers that are products of exactly four distinct primes (or tetraprimes).

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