cp's OEIS Frontend

A344093 a(n) is the smallest positive integer not already in the sequence such that a(n) + a(n-1) is the product of two distinct primes, with a(1) = 1.

%I A344093 #29 Oct 02 2021 08:09:22
%S A344093 1,5,9,6,4,2,8,7,3,11,10,12,14,19,15,18,16,17,21,13,20,26,25,30,27,24,
%T A344093 22,29,28,23,32,33,36,38,31,34,35,39,43,42,40,37,45,41,44,47,46,48,58,
%U A344093 53,62,49,57,54,52,59,56,50,61,68,51,55,60,63,66,67,74,69,64
%N A344093 a(n) is the smallest positive integer not already in the sequence such that a(n) + a(n-1) is the product of two distinct primes, with a(1) = 1.
%C A344093 This sequence is not to be confused with A243625 (similar but with "semiprime" instead of "product of two distinct primes"). This sequence omits squares of primes, whereas semiprimes include them. This sequence is also similar to A055625, in which sums of consecutive terms are primes instead of semiprimes.
%C A344093 Interestingly, the first 9 terms are a permutation of the first 9 positive integers. This is also true for the first 12, 21, 30, and 48 terms, and possibly higher values as well. This suggests that every positive integer occurs, but this is unproved.
%e A344093 a(4) = 6 because 6 is the smallest k such that a(3) + k is the product of two distinct primes.
%t A344093 a[1]=1;a[n_]:=a[n]=(k=1;While[MemberQ[Array[a,n-1],k]||Last/@FactorInteger[a[n-1]+k]!={1,1},k++];k);Array[a,100] (* _Giorgos Kalogeropoulos_, Aug 16 2021 *)
%o A344093 (Python)
%o A344093 terms = [1]
%o A344093 previous = 1
%o A344093 def isValid(num):
%o A344093   counter = 0
%o A344093   for possibleDiv in range(1, int(math.sqrt(num)) + 1):
%o A344093     if num % possibleDiv == 0:
%o A344093       counter += 1
%o A344093       if num/possibleDiv % possibleDiv == 0 and possibleDiv != 1:
%o A344093         return False
%o A344093     if counter > 2:
%o A344093       return False
%o A344093   if counter == 2:
%o A344093     return True
%o A344093   return False
%o A344093 def generateSequence(numOfTerms):
%o A344093   for i in range(numOfTerms):
%o A344093     testNum = 1
%o A344093     valid = False
%o A344093     while not valid:
%o A344093       if testNum not in terms:
%o A344093         possibleNum = previous + testNum
%o A344093         if isValid(num):
%o A344093           valid = True
%o A344093           terms.append(testNum)
%o A344093           previous = testNum
%o A344093       testNum += 1
%Y A344093 Cf. A243625, A055625, A006881.
%K A344093 nonn
%O A344093 1,2
%A A344093 _Atticus Stewart_, Aug 15 2021