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

1,2

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.

Links

Table of n, a(n) for n=1..69.

Example

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

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 *)

Prog

(Python)
import math
def isValid(num):
  counter = 0
  for possibleDiv in range(1, math.isqrt(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):
  terms = [1]
  for i in range(numOfTerms):
    testNum = 1
    while True:
      if testNum not in terms:
        if isValid(terms[-1] + testNum):
          terms.append(testNum)
          break
      testNum += 1
  return terms

Crossrefs

Cf. A243625, A055625, A006881.

Sequence in context: A303983 A388828 A073003 * A087498 A274633 A238200

Adjacent sequences: A344090 A344091 A344092 * A344094 A344095 A344096

Keyword

nonn

Author

Atticus Stewart, Aug 15 2021

Status

approved