A382453 Lexicographically earliest sequence of distinct terms such that no term is a substring of the sum of any two terms.

1, 3, 21, 23, 25, 39, 41, 43, 45, 47, 49, 221, 223, 241, 243, 2001, 2003, 2021, 2023, 2025, 2039, 2041, 2043, 2045, 2047, 2049, 2221, 2223, 2241, 2243, 2601, 2603, 2621, 2623, 2639, 2641, 2643, 2645, 4001, 4003, 4021, 4023, 4025, 4039, 4041, 4043, 4045, 4047

Offset: 1

Author

Dominic McCarty, Mar 26 2025

Keywords

• nonn
• base

Comments

The "two terms" mentioned in the name are not necessarily distinct, so no term is the substring of the double of any term.

The first digit of every term is less than 5.

Examples

			For calculating a(3):
If 4 was in the sequence, 1+3 = 4 would have 4, itself, as a substring, so it is disallowed.
If 5 was in the sequence, 5+5 = 10 would have 1 as a substring, so it is disallowed.
The first term that is allowed is 21, since 21 is the first term not to have 1, 3, or itself as a substring of any of the following: 1+21 = 22, 3+21 = 24, 21+21 = 42.
So, a(3) = 21.
		

Crossrefs

Cf. A381242.

Programs

Python

a = [1]
while len(a) < 20:
    a.append(a[-1]+1)
    while any(any(str(k) in str(a[i]+a[j]) for k in a) for i in range(len(a)) for j in range(i,len(a))): a[-1] += 1
print(a)

Python

from itertools import count, islice
def agen():    # generator of terms
    slst, alst, an = [], [], 1
    S = ["2"]  # strings of sums of two terms (including self sums)
    while True:
        alst.append(an)
        slst.append(str(an))
        yield an
        for k in count(an+1):
            sk = str(k)
            if any(sk in s for s in S): continue
            Pk = [str(ai+k) for ai in alst] + [str(k+k)]
            if any(si in s for s in Pk for si in slst+[sk]): continue
            an = k
            S += Pk
            break
print(list(islice(agen(), 48))) # Michael S. Branicky, Mar 26 2025