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A350598
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Composite numbers d such that the period k of the repetend of 1/d is > 1 and divides d-1, and d is the first such composite with a given repetend.
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1
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33, 55, 91, 99, 148, 165, 175, 246, 259, 275, 325, 370, 385, 451, 481, 495, 496, 505, 561, 592, 656, 657, 703, 715, 825, 909, 925, 1035, 1045, 1105, 1233, 1375, 1476, 1626, 1729, 1825, 1912, 2035, 2120, 2275, 2368, 2409, 2465, 2475, 2525, 2556, 2752, 2821, 2981, 3160
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OFFSET
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1,1
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COMMENTS
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This is a subset of sequence A351396, and adds the extra condition that d is included only if it is the smallest value of d with a given repetend; thus duplicate repetends are not permitted. This eliminates some values of A351396 which are powers of 10 of d. For example, 1480 is excluded because although its period (k=3 based on a repetend of 675) divides evenly into 1479, this repetend already exists for a smaller value of d, namely 148, and 3 also divides evenly into 147. 1480 is the smallest such value of d from A351396 that will be excluded based on this modification. Other values of A351396 that are excluded include 3700, 5920, 9250, 14800, ...
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LINKS
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EXAMPLE
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33 is a term since 1/33 = 0.030303..., its repetend is 03, so its period is 2, 2 divides 33-1 evenly, and there is no smaller value of d with this repetend.
148 is in the sequence because 1/148 has 675 as its repetend, so its period is 3 and 3 divides 148-1).
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PROG
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(Python)
from itertools import count, islice
from sympy import n_order, multiplicity, isprime
def A350598_gen(): # generator of terms
pset = set()
for d in count(1):
if not isprime(d):
m2, m5 = multiplicity(2, d), multiplicity(5, d)
r = max(m2, m5)
k, m = 10**r, 10**(t := n_order(10, d//2**m2//5**m5))-1
c = k//d
s = str(m*k//d-c*m).zfill(t)
if not (t <= 1 or (d-1) % t or s in pset):
yield d
pset.add(s)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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