A355636 a(1) = a(2) = 1; 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) but does not equal the sum.

1, 1, 3, 9, 18, 6, 30, 100, 24, 12, 196, 48, 20, 28, 80, 60, 72, 84, 90, 40, 42, 8, 32, 54, 10, 729, 2, 14, 81, 15, 108, 21, 22, 5, 26, 7, 27, 33, 96, 34, 56, 126, 66, 320, 35, 38, 11, 4, 39, 13, 44, 46, 132, 51, 55, 57, 162, 58, 140, 150, 70, 156, 62, 65, 17, 69, 74, 77, 19, 160, 23, 82, 78

Offset

1,3

Comments

In the first 250000 terms the smallest numbers that have not appeared are 64, 1024, 11664, 15625. It is unknown if these and all other numbers eventually appear.

See A355637 for the fixed points.

Links

Table of n, a(n) for n=1..73.

Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.

Example

a(6) = 6 as a(4) + a(5) = 9 + 18 = 27 which has four divisors, and 6 is the smallest unused number that does not equal 27 and has four divisors.

Prog

(PARI) listm(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 1; my(m = Map()); mapput(m, 1, 1); for (n=3, nn, my(s=va[n-2]+va[n-1], d=numdiv(s), k=1, vs=Vec(va, n-1)); while (mapisdefined(m, k) || (k==s) || (numdiv(k)!=d), k++); va[n] = k; mapput(m, k, n); ); va; \\ Michel Marcus, Jul 11 2022
(Python)
from sympy import divisor_count
from itertools import count, islice
def agen():
    anm1, an, mink, seen = 1, 1, 2, {1}
    yield 1
    for n in count(2):
        yield an
        k, target, tsum = mink, divisor_count(anm1+an), anm1+an
        while k in seen or k == tsum or divisor_count(k) != target: k += 1
        while mink in seen: mink += 1
        anm1, an = an, k
        seen.add(an)
print(list(islice(agen(), 73))) # Michael S. Branicky, Jul 26 2022

Crossrefs

Cf. A355637, A000005, A351001, A352768, A352867, A352774.

Sequence in context: A009216 A103072 A109228 * A103162 A310331 A261952

Adjacent sequences: A355633 A355634 A355635 * A355637 A355638 A355639

Keyword

nonn

Author

Scott R. Shannon, Jul 11 2022

Status

approved