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.
A379248(1169) = 41, A379248(1175) = 43, with a difference in indices of 6. Worth noting is the values of the terms in this, and similar, ranges: . . A379248(1167) = 943 = 23*41 , the lowest unseen multiple of 23. A379248(1168) = 1681 = 41^2. A379248(1169) = 41. A379248(1170) = 3362 = 2*41^2 , which shows the pattern of p^2 -> p -> 2*p^2. A379248(1171) = 697 = 17*41 , the lowest unseen multiple of 17. A379248(1172) = 2023 = 7*17^2 , the lowest unseen multiple of 17^2. A379248(1173) = 731 = 17*43, the lowest unseen multiple of 17. A379248(1174) = 1849 = 43^2. A379248(1175) = 43. . .
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. 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.
from sympy import factorint
from itertools import islice
from collections import Counter
fcache = dict()
def myfactors(n):
global fcache
if n in fcache: return fcache[n]
ans = Counter({p:e for p, e in factorint(n).items()})
fcache[n] = ans
return ans
def agen(): # generator of terms
yield 1
an, a, m = 2, {1, 2}, 3
while True:
yield an
k, fan = m-1, myfactors(an)
sfan = set(fan)
while True:
k += 1
if k in a: continue
fk = myfactors(k)
sfk = set(fk)
if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfk | sfan):
an = k
break
a.add(an)
print(list(islice(agen(), 88))) # Michael S. Branicky, Jan 05 2025
a(14) = 44 as 44 = 2^2*11^1, and a(13) = 14 = 2*7 which contains 2^1 as a factor, whose power differs by one from 2^2, while not containing any power of 11. This is the smallest unused number satisfying these criteria. Note that 36 = 2^2*3^2 cannot be chosen as a(13) contains no power of 3 - this is the first term to differ from A379441.
from sympy import factorint
from itertools import islice
from collections import Counter
fcache = dict()
def myfactors(n):
global fcache
if n in fcache: return fcache[n]
ans = Counter({p:e for p, e in factorint(n).items()})
fcache[n] = ans
return ans
def agen(): # generator of terms
yield 1
an, a, m = 2, {1, 2}, 3
while True:
yield an
k, fan = m-1, myfactors(an)
sfan = set(fan)
while True:
k += 1
if k in a: continue
fk = myfactors(k)
sfk = set(fk)
if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfk):
an = k
break
a.add(an)
print(list(islice(agen(), 87))) # Michael S. Branicky, May 25 2025
a(14) = 36 as 36 = 2^2*3^2 while a(13) = 14 = 2*7 which contains 2^1 as a factor, whose power differs by one from 2^2, and 7^1 as a factor, and 36 contains no power of 7. This is the smallest unused number satisfying these criteria. This is the first term to differ from A379440.
from sympy import factorint
from itertools import islice
from collections import Counter
fcache = dict()
def myfactors(n):
global fcache
if n in fcache: return fcache[n]
ans = Counter({p:e for p, e in factorint(n).items()})
fcache[n] = ans
return ans
def agen(): # generator of terms
yield 1
an, a, m = 2, {1, 2}, 3
while True:
yield an
k, fan = m-1, myfactors(an)
sfan = set(fan)
while True:
k += 1
if k in a: continue
fk = myfactors(k)
sfk = set(fk)
if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfan):
an = k
break
a.add(an)
print(list(islice(agen(), 88))) # Michael S. Branicky, May 25 2025
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