A370499 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)), Omega(a(n)) does not equal Omega(a(n-1)), the string length of a(n) does not equal the string length of a(n-1), and a(n) has no digit in common with a(n-1), where sopfr(k) is the sum of the primes dividing k, with repetition.

1, 20, 7, 15, 208, 5, 12, 305, 17, 4, 11, 6, 13, 8, 19, 200, 3, 10, 223, 9, 23, 100, 27, 101, 22, 103, 24, 107, 25, 104, 29, 105, 2, 45, 109, 26, 113, 28, 111, 40, 117, 32, 147, 38, 127, 30, 149, 33, 112, 37, 102, 43, 106, 47, 108, 35, 124, 39, 116, 49, 125, 34, 151, 36, 157, 42, 131, 44, 115

Offset: 1

Author

Scott R. Shannon, Feb 20 2024

Keywords

• nonn
• base

Comments

The fixed points begin 1, 2, 11, 13, 40, 357, 2353, 2393, 2465, 2473, 2529, 2649, 2767. It is unknown if the sequence is infinite.

Examples

			a(5) = 208 as a(4) = 15 and 208 is the smallest unused number that is coprime to 15, sopfr(208) = 21 is coprime to sopfr(15) = 8, Omega(208) = 5 does not equal Omega(15) = 2, the string length of "208" = 3 does not equal the string length of "15" = 2, and 208 has no digit in common with 15.
		

Links

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

Crossrefs

Cf. A001414, A001222, A370496, A370047, A370497, A064413, A277272, A027748.

Programs

Python

from math import gcd
from sympy import factorint
from functools import cache
from itertools import count, islice
@cache
def sWd(n):
    f = factorint(n)
    return (sum(p*e for p,e in f.items()), sum(f.values()), str(n))
def agen(): # generator of terms
    yield 1
    aset, an, mink = {1, 20}, 20, 2
    while True:
        yield an
        s, W, d = sWd(an)
        an = next(k for k, sk, Wk, dk in ((k,)+sWd(k) for k in count(mink)) if k not in aset and gcd(k, an)==1 and gcd(sk, s)==1 and Wk!=W and len(dk)!=len(d) and set(dk)&set(d)==set())
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Feb 21 2024