A repunit is a number consisting of copies of the single digit 1. The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors.
In base-10, repunits have the form
Repunits Demlo
numbers , A002275
and A002477 ).
The number of factors 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 base of the form
(3)
This gives the special cases summarized in the following table.
The idea of repunits can also be extended to negative bases. Except for requiring
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 Cunningham Number ,
Demlo Number ,
Fermat Number ,
Mersenne
Number ,
Repdigit ,
Repunit
Prime ,
Smith Number
Explore with Wolfram|Alpha
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 +/-1, b =2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev.
ed. 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 ,
A002477 /M5386, A002450 /M3914,
A002452 /M4733, A003462 /M3463,
A007583 , A007658 ,
A003463 /M4209, A003464 /M4425,
A004023 /M2114, A004061 /M2620,
A004062 /M0861, A004063 /M3836,
A004064 /M0744, A005808 /M5032,
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
therefore have exactly 
decimal digits. Amazingly, the squares of the repunits 
give the Demlo
numbers, 
,
, 
, ... (OEIS A002275
and A002477).
, 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS
A046053).
, giving a base-
repunit as number of the form
to be odd, the math is very similar (Dubner
and Granlund 2000).
-repunits
