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 therefore have exactly decimal digits. Amazingly, the squares of the repunits give the Demlo
numbers , ,
, , ... (OEIS A002275
and A002477 ).
The number of factors for the base-10 repunits for , 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS
A046053 ).
A repunit that is a prime number is known as a repunit prime .
Repunits can be generalized to base , giving a base- repunit as number 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 to be odd, the math is very similar (Dubner
and Granlund 2000).
OEIS -repunitsA066443 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
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References Beiler, A. H. "11111...111." Ch. 11 in Recreations
in the Theory of Numbers: The Queen of Mathematics Entertains. New York:
Dover, 1966. 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. Providence, RI: Amer. Math. Soc., 1988. Dudeney, H. E.
The
Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson
and Sons, 1949. Gardner, M. The
Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University
of Chicago Press, pp. 85-86, 1984. Granlund, T. "Repunits."
http://www.swox.com/gmp/repunit.html . Madachy,
J. S. Madachy's
Mathematical Recreations. New York: Dover, pp. 152-153, 1979. Ribenboim,
P. "Repunits and Similar Numbers." §5.5 in The
New Book of Prime Number Records. New York: Springer-Verlag, pp. 350-354,
1996. 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 --A Wolfram Web Resource. https://mathworld.wolfram.com/Repunit.html
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