A repunit is a positive integer consisting of one or more copies of the single digit 1. The term "repunit"
was coined by Beiler (1966), who also gave the first tabulation of known factors.
For a positive integer base -10 repunits have the form
Repunits decimal digits .
The squares of the repunits Demlo numbers ,
A002275
and A002477 ).
The numbers of prime factors counted with multiplicity for the base -10 repunits for A046053 ).
A repunit that is a prime number is known as a repunit prime .
Repunits can be generalized to an integer base base -of
the form
(3)
This gives the special cases summarized in the following table.
The same formula gives an all-ones numeral in a negative integer base iff odd
(Dubner and Granlund 2000). The following table lists the first few positive values
for several bases .
OEIS A066443 1, 7, 61, 547, 4921, 44287, 398581, ... A007583 1, 3, 11, 43, 171, 683, 2731, ... 2 A000225 1,
3, 7, 15, 31, 63, 127, ... 3 A003462 1, 4, 13, 40, 121,
364, ... 4 A002450 1, 5, 21, 85, 341, 1365, ... 5 A003463 1,
6, 31, 156, 781, 3906, ... 6 A003464 1, 7, 43, 259, 1555,
9331, ... 7 A023000 1, 8, 57, 400, 2801, 19608, ... 8 A023001 1,
9, 73, 585, 4681, 37449, ... 9 A002452 1, 10, 91, 820, 7381,
66430, ... 10 A002275 1, 11, 111, 1111, 11111, ... 11 A016123 1,
12, 133, 1464, 16105, 177156, ... 12 A016125 1,
13, 157, 1885, 22621, 271453, ...
See also Binary Expansion ,
Cunningham Number ,
Demlo Number ,
Fermat
Number ,
Mersenne Number ,
Repdigit ,
Repunit Prime ,
Smith
Number
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References Beiler, A. H. "11111...111." Ch. 11 in Recreations
in the Theory of Numbers: The Queen of Mathematics Entertains. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff,
S. S. Jr.; and Tuckerman, B. Factorizations
of b n b =2, 3, 5, 6, 7, 10, 11, 12 Up to High
Powers, rev. ed. Providence, RI: Amer. Math. Soc., 1988. Dubner,
H. and Granlund, T. "Primes of the Form J. Integer Seq. 3 , Article
00.2.7, 2000. https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html . Dudeney,
H. E. The
Canterbury Puzzles and Other Curious Problems, 7th ed. Gardner, M. The
Sixth Book of Mathematical Games from Scientific American. Granlund, T. "Repunits."
http://www.swox.com/gmp/repunit.html . Madachy,
J. S. Madachy's
Mathematical Recreations. Ribenboim,
P. "Repunits and Similar Numbers." §5.5 in The
New Book of Prime Number Records. Sloane, N. J. A. Sequences A000043 /M0672,
A000225 /M2655, A000978 ,
A001562 , A002275 ,
A002450 /M3914, A002452 /M4733,
A002477 /M5386, A003462 /M3463,
A003463 /M4209, A003464 /M4425,
A004023 /M2114, A004061 /M2620,
A004062 /M0861, A004063 /M3836,
A004064 /M0744, A005808 /M5032,
A007583 , A007658 ,
A016123 , A016125 ,
A023000 , A023001 ,
A028491 /M2643, A046053 ,
A057171 , A057172 ,
A057173 , A057175 ,
A057177 , A057178 ,
A066443 , and A084740
in "The On-Line Encyclopedia of Integer Sequences." Yates,
S. "Peculiar Properties of Repunits." J. Recr. Math. 2 , 139-146,
1969. Yates, S. "The Mystique of Repunits." Math. Mag. 51 ,
22-28, 1978. Yates, S. Repunits and Reptends. Delray Beach, FL:
S. Yates, 1982. Referenced on Wolfram|Alpha Repunit
Cite this as:
Weisstein, Eric W. "Repunit." From MathWorld https://mathworld.wolfram.com/Repunit.html
Subject classifications
, base-10 repunits have the form
therefore have exactly 
decimal digits.
The squares of the repunits 
give the Demlo numbers,
, 
, 
, ... (OEIS A002275
and A002477).
, 2, ... are 0, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS
A046053).
, giving a base-
repunit of
the form
. Its value is greater than 0 iff 
is odd
(Dubner and Granlund 2000). The following table lists the first few positive values
for several bases.
-repunits
." J. Integer Seq. 3, Article
00.2.7, 2000.