A379442 - OEIS

%I #17 Jan 05 2025 10:24:13

%S 1,2,4,6,9,3,18,12,8,16,24,20,14,44,10,25,5,50,15,63,27,45,21,49,7,98,

%T 28,22,52,30,36,42,68,26,60,34,76,40,48,32,64,96,80,56,92,38,84,46,

%U 116,62,132,58,124,66,117,33,90,39,99,51,126,57,153,54,81,135,162,108,72,156,70,75,35,147,77,121,11,242,55,150,65,169,13,338,91,245,119,289

%N a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) such that the exponents of each distinct prime factor of a(n-1) differ by one from those of the same prime factors of a(n), while the exponents of each distinct prime factor of a(n) differ by one from those of the same prime factors of a(n-1).

%C The sequence shows similar behavior to A379248 - prime terms p are proceeded by p^2 and followed by 2*p^2, primes appear in their natural order, primes can be divisors of terms long before they appear as a term themselves, there are long runs of prime terms that are separated by six terms, and prime terms appear when the terms overall go through intermittent periods of large oscillations in value.

%C The most significant difference is the terms are concentrated along two different lines when between the periods of large oscillation. These appear to be comprised of terms that jump between values of 2*k and 2^2*k' or 3*k and 3^2*k', with k,k'>1. Sometimes between these lines are successive terms comprised of multiples of large powers of 2 or 3; see the attached image.

%C In the first 100000 terms there are eleven fixed points. However, as the regions of oscillating terms crosses the a(n) = n line it is possible more exist for larger values of n.

%C The sequence is conjectured to be a permutation of the positive integers.

%H Scott R. Shannon, <a href="/A379442/b379442.txt">Table of n, a(n) for n = 1..20000</a>

%H Scott R. Shannon, <a href="/A379442/a379442.png">Colored image of the first 20000 terms</a>. The terms with one, two, three,... as their maximum prime factorization exponent are colored red, orange, yellow,..., violet, with white terms having exponents > 8. The green line is a(n) = n.

%e a(3) = 4 as 4 is unused and shares a factor with a(2) = 2, while 4 = 2^2 which has 2 as the exponent of the prime 2, while a(2) = 2^1 which has exponent 1. As these exponents differ by one, 4 is acceptable.

%e a(5) = 9 as 9 is unused and shares a factor with a(4) = 6, while 9 = 3^2 which has 2 as the exponent of the prime 3 and exponent 0 for the prime 2, while a(4) = 2^1*3^1 which has exponent 1 for both primes 2 and 3. As these both differ by one, 9 is acceptable. Note that although 8 shares a factor with 6, 8 = 2^3 which has exponent 3 for the prime 2, and as that does not differ by one from the exponent 1 for the prime 2 in 6, 8 cannot be chosen. This is the first term to differ from A379248.

%o (Python)

%o from sympy import factorint

%o from itertools import islice

%o from collections import Counter

%o fcache = dict()

%o def myfactors(n):

%o     global fcache

%o     if n in fcache: return fcache[n]

%o     ans = Counter({p:e for p, e in factorint(n).items()})

%o     fcache[n] = ans

%o     return ans

%o def agen(): # generator of terms

%o     yield 1

%o     an, a, m = 2, {1, 2}, 3

%o     while True:

%o         yield an

%o         k, fan = m-1, myfactors(an)

%o         sfan = set(fan)

%o         while True:

%o             k += 1

%o             if k in a: continue

%o             fk = myfactors(k)

%o             sfk = set(fk)

%o             if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfk | sfan):

%o                 an = k

%o                 break

%o         a.add(an)

%o print(list(islice(agen(), 88))) # _Michael S. Branicky_, Jan 05 2025

%Y Cf. A379440, A379441, A379248, A124010, A027746, A051903, A064413, A348086.

%Y Cf. A379557 (fixed points), A379558 (index where prime n appears as a term), A379559 (index where n appears as a term).

%K nonn,new

%O 1,2

%A _Scott R. Shannon_, Dec 23 2024