This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379442 #21 May 25 2025 16:07:21
%S A379442 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 A379442 28,22,52,30,36,42,68,26,60,34,76,40,48,32,64,96,80,56,92,38,84,46,
%U A379442 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 A379442 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 A379442 The sequence shows similar behavior to A379248 - prime terms p are preceded 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 A379442 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 A379442 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 A379442 The sequence is conjectured to be a permutation of the positive integers.
%H A379442 Scott R. Shannon, <a href="/A379442/b379442.txt">Table of n, a(n) for n = 1..20000</a>
%H A379442 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 A379442 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 A379442 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 A379442 (Python)
%o A379442 from sympy import factorint
%o A379442 from itertools import islice
%o A379442 from collections import Counter
%o A379442 fcache = dict()
%o A379442 def myfactors(n):
%o A379442 global fcache
%o A379442 if n in fcache: return fcache[n]
%o A379442 ans = Counter({p:e for p, e in factorint(n).items()})
%o A379442 fcache[n] = ans
%o A379442 return ans
%o A379442 def agen(): # generator of terms
%o A379442 yield 1
%o A379442 an, a, m = 2, {1, 2}, 3
%o A379442 while True:
%o A379442 yield an
%o A379442 k, fan = m-1, myfactors(an)
%o A379442 sfan = set(fan)
%o A379442 while True:
%o A379442 k += 1
%o A379442 if k in a: continue
%o A379442 fk = myfactors(k)
%o A379442 sfk = set(fk)
%o A379442 if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfk | sfan):
%o A379442 an = k
%o A379442 break
%o A379442 a.add(an)
%o A379442 print(list(islice(agen(), 88))) # _Michael S. Branicky_, Jan 05 2025
%Y A379442 Cf. A379440, A379441, A379248, A124010, A027746, A051903, A064413, A348086.
%Y A379442 Cf. A379557 (fixed points), A379558 (index where prime n appears as a term), A379559 (index where n appears as a term).
%K A379442 nonn
%O A379442 1,2
%A A379442 _Scott R. Shannon_, Dec 23 2024