A370497 a(1) = 1; for n > 1, a(n) is smallest unused number such that a(n) is coprime to a(n-1), sopfr(a(n)) is coprime to sopfr(a(n-1)), and Omega(a(n)) does not equal Omega(a(n-1)), where sopfr(k) is the sum of the primes dividing k, with repetition.

1, 2, 27, 4, 3, 10, 11, 6, 7, 8, 5, 9, 13, 12, 17, 14, 19, 15, 23, 16, 29, 18, 31, 20, 21, 37, 22, 41, 24, 25, 28, 33, 40, 39, 43, 26, 45, 32, 47, 30, 53, 34, 59, 35, 48, 49, 44, 57, 52, 51, 56, 55, 54, 61, 36, 67, 38, 63, 46, 71, 42, 73, 50, 79, 58, 75, 62, 83, 60, 89, 64, 97, 65, 76, 69, 68

Offset: 1

Author

Scott R. Shannon, Feb 20 2024

Keywords

• nonn

Comments

In the first 100000 terms the primes 11, 7 and 5 appear in reverse order, but all others appear in their natural order. In the same range the fixed points begin 1, 2, 4, 13, 294, 295, 296, 299, 304, 309, 640, 649. The sequence is conjectured to be a permutation of the positive numbers.

Examples

			a(3) = 27 as a(2) = 2 and 27 is the smallest unused number that is coprime to 2, sopfr(27) = 9 is coprime to sopfr(2) = 2, and Omega(27) = 3 does not equal Omega(2) = 1.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A001222, A370496, A370047, A064413, A277272, A027748.

Programs

Python

from math import gcd
from sympy import factorint
from functools import cache
from itertools import count, islice
@cache
def sW(n):
    f = factorint(n)
    return (sum(p*e for p,e in f.items()), sum(f.values()))
def agen(): # generator of terms
    yield 1
    aset, an, mink = {1, 2}, 2, 3
    while True:
        yield an
        s, W = sW(an)
        an = next(k for k, sk, Wk in ((k,)+sW(k) for k in count(mink)) if k not in aset and gcd(k, an)==1 and gcd(sk, s)==1 and Wk!=W)
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 76))) # Michael S. Branicky, Feb 21 2024