A390396 Least positive integer not the sum, product or power of any previous pair of distinct terms.

1, 2, 4, 7, 10, 13, 18, 21, 24, 27, 30, 33, 38, 41, 44, 47, 50, 53, 58, 61, 64, 67, 73, 78, 81, 87, 90, 93, 98, 101, 104, 107, 110, 113, 118, 121, 124, 127, 133, 138, 141, 144, 149, 155, 158, 161, 172, 175, 178, 181, 184, 192, 195, 198, 201, 204, 209, 215, 218, 221, 224, 229, 235

Offset

1,2

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Formula

a(n+1) > a(n) for all n > 1 since 1 is a term and hence a(n) is in the set of products and a(n+1) >= a(n) by construction. - Michael S. Branicky, Nov 13 2025

Prog

(Python)
from itertools import combinations
def generate_sequence(n_terms):
    sequence = [1]
    while len(sequence) < n_terms:
        sums = set()
        prods = set()
        powers = set()
        for a, b in combinations(sequence, 2):
            sums.add(a + b)
            prods.add(a * b)
            try:
                powers.add(a ** b)
            except OverflowError:
                pass
            try:
                powers.add(b ** a)
            except OverflowError:
                pass
        forbidden = sums.union(prods).union(powers)
        candidate = 1
        while candidate in sequence or candidate in forbidden:
            candidate += 1
        sequence.append(candidate)
    return sequence
seq = generate_sequence(15)
(Python)
from math import isqrt
from itertools import count
def aupto(limit):
    alst, an, aset, sums, prds, pows, r = [], 1, {1}, {1}, {1}, set(), isqrt(limit)+1
    while an <= limit:
        alst.append(an)
        aset.add(an)
        an = next(k for k in count(an+1) if all(k not in S for S in [sums, prds, pows]))
        sums |= {t for s in aset if (t:= an+s) <= limit}
        prds |= {t for p in aset if (t:= an*p) <= limit}
        for p in aset:
            if an > r or (t:=an**p) > limit: break
            pows.add(t)
        for p in aset:
            if p > r or (t:=p**an) > limit: break
            pows.add(t)
    return alst
print(aupto(300)) # Michael S. Branicky, Nov 08 2025

Crossrefs

Cf. A066512.

Sequence in context: A194172 A391537 A173537 * A049983 A168112 A170894

Adjacent sequences: A390393 A390394 A390395 * A390397 A390398 A390399

Keyword

nonn

Author

Marc Morgenegg, Nov 04 2025

Status

approved