A355647 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has the same number of divisors as the sum a(n-2) + a(n-1).

1, 2, 3, 5, 6, 7, 11, 12, 13, 4, 17, 8, 9, 19, 18, 23, 29, 20, 25, 28, 31, 37, 32, 10, 24, 14, 15, 41, 30, 43, 47, 60, 53, 59, 48, 61, 67, 40, 71, 21, 44, 22, 42, 64, 26, 72, 45, 50, 27, 33, 84, 52, 54, 34, 56, 90, 35, 38, 73, 39, 80, 46, 96, 51, 63, 66, 55, 49, 70, 57, 79, 78, 83, 58, 62, 120

Offset

1,2

Comments

In the first 500000 terms the smallest numbers that have not appeared are 15625, 25600, 28561, 36864. It is unknown if these and all other numbers eventually appear. In the same range on eighty-two occasions a(n) equals the sum of the previous two terms, these values begin 3, 5, 17, 64, 90, 73, 120, 144, 192.

See A355648 for the fixed points.

Links

Table of n, a(n) for n=1..76.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Example

a(5) = 6 as a(3) + a(4) = 3 + 5 = 8 which has four divisors, and 6 is the smallest unused number that has four divisors.

Prog

(Python)
from sympy import divisor_count
from itertools import count, islice
def agen():
    anm1, an, mink, seen = 1, 2, 3, {1, 2}
    yield 1
    for n in count(2):
        yield an
        k, target = mink, divisor_count(anm1+an)
        while k in seen or divisor_count(k) != target: k += 1
        while mink in seen: mink += 1
        anm1, an = an, k
        seen.add(an)
print(list(islice(agen(), 76))) # Michael S. Branicky, Jul 26 2022

Crossrefs

Cf. A355648, A355636, A000005, A351001, A352768, A352867, A352774.

Sequence in context: A353954 A059041 A129128 * A367585 A164922 A205523

Adjacent sequences: A355644 A355645 A355646 * A355648 A355649 A355650

Keyword

nonn

Author

Scott R. Shannon, Jul 12 2022

Status

approved