A359065 Lexicographically earliest sequence of distinct positive composite integers such that no subsequence sums to a prime and in which all terms are coprime.

4, 21, 65, 209, 391, 3149, 9991, 368131, 57556589, 14865154981

Offset: 1

Author

Conor Houghton, Dec 15 2022

Keywords

• nonn
• more

Comments

The sequences A052349 and A068638 are composed of integers starting from one with the rule that no subsequence has a prime sum; these sequences start with one. Starting with a different number seems to result in straightforward geometric sequences, for example the sequence with no prime subsequence sums starting with four is 4, 6, 8, and so on. One way to avoid this is to enforce a coprime rule, requiring that the entries to the sequence are coprime. It is not clear whether the sequence is infinite.

Crossrefs

Cf. A052349, A068638.

Programs

Mathematica

k = 4; K = {k}; f = {2}; q = Subsets[K]; While[Length@K < 10, k++; If[! PrimeQ[k] && ! IntersectingQ[FactorInteger[k][[All, 1]], f], s = k; z = 0; For[p = 1, p <= Length@q, p++, If[PrimeQ[Total[q[[p]]] + k], z = 1; Break[]]]; If[z == 0, AppendTo[K, k]; q = Subsets[K]; AppendTo[f, FactorInteger[k][[All, 1]]]; f = Flatten[f]]]]; Print[K] (* Samuel Harkness, Apr 11 2023 *)

Python

import sys
import math
from sympy.ntheory import primefactors
from sympy.ntheory import primerange
def intersection(lst1, lst2):
    lst3 = [value for value in lst1 if value in lst2]
    return len(lst3)
n_primes=1000000
factors=[primefactors(n) for n in range(0,n_primes)]
primes=list(primerange(0, n_primes))
sequence=[4]
sums=[sequence[0]]
prime_factors=[f for f in factors[sequence[0]]]
big_n=8
while len(sequence)
				

Python

from math import gcd
from sympy import isprime
from itertools import islice
def agen(start=4): # generator of terms
    alst, k, sums = [start], start+1, {0} | {start}
    while True:
        yield alst[-1]
        while any(gcd(k, an) != 1 for an in alst) or \
              any(k+s not in sums and isprime(k+s) for s in sums):
            k += 1
        alst.append(k)
        sums.update([k + s for s in sums])
        k += 1
print(list(islice(agen(), 8))) # Michael S. Branicky, Dec 16 2022

Extensions

a(8)-a(9) from Michael S. Branicky, Dec 15 2022

a(10) from Rémy Sigrist, Dec 16 2022