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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
LINKS
EXAMPLE
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.
PROG
(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
CROSSREFS
Sequence in context: A076594 A174991 A217418 * A196102 A196099 A102409
KEYWORD
nonn,base
AUTHOR
Scott R. Shannon, Feb 20 2024
STATUS
approved

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Last modified June 22 12:33 EDT 2024. Contains 373571 sequences. (Running on oeis4.)