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.

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, 22, 29, 28, 23, 32, 33, 36, 38, 31, 34, 35, 39, 43, 42, 40, 37, 45, 41, 44, 47, 46, 48, 58, 53, 62, 49, 57, 54, 52, 59, 56, 50, 61, 68, 51, 55, 60, 63, 66, 67, 74, 69, 64

Offset: 1

Author

Atticus Stewart, Aug 15 2021

Keywords

• nonn

Comments

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.

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.

Examples

			a(4) = 6 because 6 is the smallest k such that a(3) + k is the product of two distinct primes.
		

Crossrefs

Cf. A243625, A055625, A006881.

Programs

Mathematica

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 *)

Python

terms = [1]
previous = 1
def isValid(num):
  counter = 0
  for possibleDiv in range(1, int(math.sqrt(num)) + 1):
    if num % possibleDiv == 0:
      counter += 1
      if num/possibleDiv % possibleDiv == 0 and possibleDiv != 1:
        return False
    if counter > 2:
      return False
  if counter == 2:
    return True
  return False
def generateSequence(numOfTerms):
  for i in range(numOfTerms):
    testNum = 1
    valid = False
    while not valid:
      if testNum not in terms:
        possibleNum = previous + testNum
        if isValid(num):
          valid = True
          terms.append(testNum)
          previous = testNum
      testNum += 1