A095370 -id:A095370 - cp's OEIS frontend

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111

Offset: 0

N. J. A. Sloane

R_n is a string of n 1's.
Base-4 representation of Jacobsthal bisection sequence A002450. E.g., a(4)= 1111 because A002450(4)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1*(4^3) + 1*(4^2) + 1*(4^1) + 1. - Paul Barry, Mar 12 2004
Except for the first two terms, these numbers cannot be perfect squares, because x^2 != 11 (mod 100). - Zak Seidov, Dec 05 2008
For n >= 0: a(n) = (A000225(n) written in base 2). - Jaroslav Krizek, Jul 27 2009, edited by M. F. Hasler, Jul 03 2020
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - Bernard Schott, Dec 24 2012
Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - Bernard Schott, Dec 24 2012
The terms 0 and 1 are the only squares in this sequence, as a(n) == 3 (mod 4) for n>=2. - Nehul Yadav, Sep 26 2013
For n>=2 the multiplicative order of 10 modulo the a(n) is n. - Robert G. Wilson v, Aug 20 2014
The above is a special case of the statement that the order of z modulo (z^n-1)/(z-1) is n, here for z=10. - Joerg Arndt, Aug 21 2014
From Peter Bala, Sep 20 2015: (Start)
Let d be a divisor of a(n). Let m*d be any multiple of d. Split the decimal expansion of m*d into 2 blocks of contiguous digits a and b, so we have m*d = 10^k*a + b for some k, where 0 <= k < number of decimal digits of m*d. Then d divides a^n - (-b)^n (see McGough). For example, 271 divides a(5) and we find 2^5 + 71^5 = 11*73*271*8291 and 27^5 + 1^5 = 2^2*7*31*61*271 are both divisible by 271. Similarly, 4*271 = 1084 and 10^5 + 84^5 = 2^5*31*47*271*331 while 108^5 + 4^5 = 2^12*7*31*61*271 are again both divisible by 271. (End)
Starting with the second term this sequence is the binary representation of the n-th iteration of the Rule 220 and 252 elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016
If p > 5 is a prime, then p divides a(p-1). - Thomas Ordowski, Apr 10 2016
0, 1 and 11 are only terms that are of the form x^2 + y^2 + z^2 where x, y, z are integers. In other words, a(n) is a member of A004215 for all n > 2. - Altug Alkan, May 08 2016
Except for the initial terms, the binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 17 2017
The term "repunit" was coined by Albert H. Beiler in 1964. - Amiram Eldar, Nov 13 2020
q-integers for q = 10. - John Keith, Apr 12 2021
Binomial transform of A001019 with leading zero. - Jules Beauchamp, Jan 04 2022

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, chapter XI, p. 83.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 235-237.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 197-198.
Samuel Yates, Peculiar Properties of Repunits, J. Recr. Math. 2, 139-146, 1969.
Samuel Yates, Prime Divisors of Repunits, J. Recr. Math. 8, 33-38, 1975.

David Wasserman, Table of n, a(n) for n = 0..1000
Eudes Antonio Costa and Fernando Soares Carvalho, On repunit polynomials sequence, Braz. Elec. J. Math. (2024). See pp. 2, 15.
Eudes Antonio Costa, Douglas Catulio Santos, Paula Maria Machado Cruz Catarino, and Elen Viviani Pereira Spreafico, On Gaussian and Quaternion Repunit Numbers, Rev. Mat. UFOP (Brazil, 2024) Vol. 2. See p. 2.
Eudes Antonio Costa, Paula Maria Machado Cruz Catarino, and Douglas Catulio Santos, A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence, Symmetry (2024) Vol. 17, No. 1, 28.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Douglas Catulio Santos, Eudes Antonio Costa, and Paula Maria Machado Cruz Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
W. M. Snyder, Factoring Repunits, Am. Math. Monthly, Vol. 89, No. 7 (1982), pp. 462-466.
Amelia Carolina Sparavigna, On Repunits, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Repunit.
Eric Weisstein's World of Mathematics, Demlo Number.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Wikipedia, Repunit.
Amin Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6.
Stephen Wolfram, A New Kind of Science.
Samuel Yates, The Mystique of Repunits, Math. Mag., Vol. 51, No. 1 (1978), pp. 22-28.
Index to Elementary Cellular Automata.
Index entries for 10-automatic sequences.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (11,-10).
Index entries for "core" sequences.

Partial sums of 10^n (A011557). Factors: A003020, A067063.
Bisections give A099814, A100706.
Cf. A000042, A002276, A002277, A002278, A002279, A002280, A002281, A002282, A002283, A004023, A046053, A059988, A065444, A075412, A075415, A083278, A095370, A102380, A125134, A178635, A204845, A204846, A204847, A204848, A206244.
Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.
Cf. A000225, A001019, A001045, A002450, A004215, A015565, A125118, A242614, A242622.
Cf. A010785, A017173, A105279.

a002275 = (`div` 9) . subtract 1 . (10 ^)
a002275_list = iterate ((+ 1) . (* 10)) 0
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

[(10^n-1)/9: n in [0..25]]; // Vincenzo Librandi, Nov 06 2014

seq((10^k - 1)/9, k=0..30); # Wesley Ivan Hurt, Sep 28 2013

Table[(10^n - 1)/9, {n, 0, 19}] (* Alonso del Arte, Nov 15 2011 *)
Join[{0},Table[FromDigits[PadRight[{},n,1]],{n,20}]] (* Harvey P. Dale, Mar 04 2012 *)

a[0]:0$
a[1]:1$
a[n]:=11*a[n-1]-10*a[n-2]$
A002275(n):=a[n]$
makelist(A002275(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=(10^n-1)/9; \\ Michael B. Porter, Oct 26 2009

my(x='x+O('x^30)); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ Altug Alkan, Apr 10 2016

print([(10**n-1)//9 for n in range(100)]) # Michael S. Branicky, Apr 30 2022

[lucas_number1(n, 11, 10) for n in range(21)]  # Zerinvary Lajos, Apr 27 2009

a(n) = 10*a(n-1) + 1, a(0)=0.
a(n) = A000042(n) for n >= 1.
Second binomial transform of Jacobsthal trisection A001045(3n)/3 (A015565). - Paul Barry, Mar 24 2004
G.f.: x/((1-10*x)*(1-x)). Regarded as base b numbers, g.f. x/((1-b*x)*(1-x)). - Franklin T. Adams-Watters, Jun 15 2006
a(n) = 11*a(n-1) - 10*a(n-2), a(0)=0, a(1)=1. - Lekraj Beedassy, Jun 07 2006
a(n) = A125118(n,9) for n>8. - Reinhard Zumkeller, Nov 21 2006
a(n) = A075412(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = A242614(n,A242622(n)). - Reinhard Zumkeller, Jul 17 2014
E.g.f.: (exp(9*x) - 1)*exp(x)/9. - Ilya Gutkovskiy, May 11 2016
a(n) = Sum_{k=0..n-1} 10^k. - Torlach Rush, Nov 03 2020
Sum_{n>=1} 1/a(n) = A065444. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Aug 02 2025: (Start)
a(n) = A002283(n)/9 = A105279(n)/10.
a(n) = A010785(A017173(n-1)) for n >= 1. (End)

A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 3, 4, 4, 4, 7, 4, 6, 6, 6, 4, 9, 5, 6, 8, 8, 4, 11, 3, 9, 9, 9, 3, 12, 7, 8, 9, 10, 7, 15, 5, 13, 8, 8, 9, 14, 5, 5, 8, 13, 6, 17, 6, 13, 12, 8, 4, 15, 6, 12, 10, 11, 6, 16, 10, 14, 8, 10, 4, 22, 9, 7, 16, 17, 9, 17, 5, 12, 8, 14, 4, 20, 5, 9, 14, 8, 10, 18

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of 11...11 (Repunit).
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): this sequence (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A002283, A046053, A085035, A003020, A046107, A005422, A061075, A102380, A095370, A147556, A081317, A081318, A102347, A112505.

PrimeOmega[10^Range[70]-1] (* Jayanta Basu, May 29 2013 *)

a(n)=bigomega(10^n-1) \\ Charles R Greathouse IV, Sep 14 2015

Mobius transform of A085035 - T. D. Noe, Jun 19 2003
a(n) = Omega(10^n -1) = Omega(R_n) + 2 = A046053(n) + 2 {where Omega(n) = A001222(n) and R_n = (10^n - 1)/9 = A002275(n)}. - Lekraj Beedassy, Jun 09 2006
a(n) = A001222(A002283(n)). - Ray Chandler, Apr 22 2017

Erroneous b-file replaced by Ray Chandler, Apr 26 2017

A046053 Total number of prime factors of the repunit R(n) = (10^n-1)/9.

Original entry on oeis.org

0, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, 4, 6, 6, 2, 9, 1, 7, 7, 7, 1, 10, 5, 6, 7, 8, 5, 13, 3, 11, 6, 6, 7, 12, 3, 3, 6, 11, 4, 15, 4, 11, 10, 6, 2, 13, 4, 10, 8, 9, 4, 14, 8, 12, 6, 8, 2, 20, 7, 5, 14, 15, 7, 15, 3, 10, 6, 12, 2, 18, 3, 7, 12, 6, 8, 16, 6, 15, 13, 7, 3, 22, 7, 8, 10, 15, 5

Offset: 1

Eric W. Weisstein

nonn

			R(6) = 111111 = (3) (7) (11) (13) (37), so a(6) = 5.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Eric Weisstein's World of Mathematics, Repunit.

Cf. A001222, A002275, A057951. For the number of distinct prime factors see A095370.

Table[PrimeOmega[(10^n - 1)/9], {n, 60}] (* Michael De Vlieger, Apr 29 2015 *)

a(n)=bigomega(10^n\9) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A001222(A002275(n)). - Ray Chandler, Apr 22 2017

a(n) = A057951(n) - 2. - Ray Chandler, Apr 24 2017

A102347 Number of distinct prime factors of 10^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 3, 5, 3, 5, 3, 7, 4, 5, 6, 7, 3, 8, 2, 8, 7, 7, 2, 10, 6, 7, 5, 9, 6, 13, 4, 12, 6, 7, 8, 11, 4, 4, 6, 12, 5, 14, 5, 11, 9, 7, 3, 13, 5, 11, 8, 10, 5, 12, 9, 13, 6, 9, 3, 20, 8, 6, 13, 16, 8, 14, 4, 11, 6, 13, 3, 17, 4, 8, 12, 7, 9, 15, 7, 16, 10, 8, 4, 21, 8, 9, 10, 15, 6, 21, 13

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 20 2005

nonn

Table of n, a(n) for n = 1..352

Cf. A001221, A002283, A070528.

A102347 := proc(n)
    10^n-1 ;
    A001221(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[PrimeNu[10^n-1],{n,100}] (* The program will take a long time to execute *) (* Harvey P. Dale, Jan 18 2015 *)

a(n) = omega(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A001221(A002283(n)) = A001221(10^n - 1).

a(n) = A001221(R_n) + (n^2 mod 3) = A095370(n) + (n^2 mod 3), where R_n = (10^n-1)/9 = A002275(n). That is, a(n) = A095370(n) for n=3k; otherwise a(n) = A095370(n) + 1. - Lekraj Beedassy, Jun 09 2006

Terms to a(280) and a(323)-a(352) in b-file from Max Alekseyev, Dec 28 2011, Apr 26 2022

a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017

A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

Original entry on oeis.org

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357

Offset: 1

Robert Munafo

Consider an m-digit number n. Square it and add the right m digits to the left m or m-1 digits. If the resultant sum is n, then n is a term of the sequence.
4879 and 5292 are in A006886 but not in this version.
Shape of plot (see links) seems to consist of line segments whose lengths along the x-axis depend on the number of unitary divisors of 10^m-1 which is equal to 2^w if m is a multiple of 3 or 2^(w+1) otherwise, where w is the number of distinct prime factors of the repunit of length m (A095370). w for m = 60 is 20, whereas w <= 15 for m < 60. This leads to the long segment corresponding to m = 60. - Chai Wah Wu, Jun 02 2016
If n*(n-1) is divisible by 10^m-1 then n is a term where m is the number of decimal digits in n. - Giorgos Kalogeropoulos, Mar 27 2025

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Chai Wah Wu, Table of n, a(n) for n = 1..50000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Douglas E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Robert Munafo, Kaprekar Sequences.
Eric Weisstein's World of Mathematics, Kaprekar Number.
Chai Wah Wu, Semilog plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Chai Wah Wu, Plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642, A095370.

Fixed points of A380585.

a053816 n = a053816_list !! (n-1)
a053816_list = 1 : filter f [4..] where
   f x = length us - length vs <= 1 &&
         read (reverse us) + read (reverse vs) == x
         where (us, vs) = splitAt (length $ show x) (reverse $ show (x^2))
-- Reinhard Zumkeller, Oct 04 2014

kapQ[n_]:=Module[{idn2=IntegerDigits[n^2],len},len=Length[idn2];FromDigits[ Take[idn2,Floor[len/2]]]+FromDigits[Take[idn2, -Ceiling[len/2]]]==n]; Select[Range[540000],kapQ] (* Harvey P. Dale, Aug 22 2011 *)
ktQ[n_] := ((x = n^2) - (z = FromDigits[Take[IntegerDigits[x], y = -IntegerLength[n]]]))*10^y + z == n; Select[Range[540000], ktQ] (* Jayanta Basu, Aug 04 2013 *)
Select[Range[540000],Total[FromDigits/@TakeDrop[IntegerDigits[#^2], Floor[ IntegerLength[ #^2]/2]]] ==#&] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jun 03 2016 *)
maxDigits=6; Flatten[Table[lst={};sub=Subsets@FactorInteger[v=10^d-1]; Do[a=Times@@Power@@@s; n=ChineseRemainder[{0,1},{a,v/a},1]; If[10^(d-1)<=n<10^d,AppendTo[lst,n]],{s,sub}];Union@lst,{d,maxDigits}]] (* Giorgos Kalogeropoulos, Mar 27 2025 *)

isok(n) = n == vecsum(divrem(n^2, 10^(1+logint(n, 10)))); \\ Ruud H.G. van Tol, Jun 02 2024

def is_A053816(n): return n==sum(divmod(n**2,10**len(str(n)))) and n
print(upto_1e5:=list(filter(is_A053816, range(10**5)))) # M. F. Hasler, Mar 28 2025

More terms from Michel ten Voorde, Apr 11 2001

A070529 Number of divisors of repunit 111...111 (with n digits).

Original entry on oeis.org

1, 2, 4, 4, 4, 32, 4, 16, 12, 16, 4, 128, 8, 16, 64, 64, 4, 384, 2, 128, 128, 96, 2, 1024, 32, 64, 64, 256, 32, 8192, 8, 2048, 64, 64, 128, 3072, 8, 8, 64, 2048, 16, 24576, 16, 1536, 768, 64, 4, 8192, 16, 1024, 256, 512, 16, 8192, 256, 4096

Offset: 1

Henry Bottomley, May 02 2002

			a(9) = 12 since the divisors of 111111111 are 1, 3, 9, 37, 111, 333, 333667, 1001001, 3003003, 12345679, 37037037, 111111111.

Table of n, a(n) for n = 1..352

Cf. A000005, A002275, A070528, A051064, A004023, A046413, A003020, A095370, A176973, A046053, A046412, A102380.

a(n) = numdiv(10^n\9); \\ Michel Marcus, Sep 09 2015

a(n) = A000005(A002275(n)).
a(n) = A070528(n)*A051064(n)/(A051064(n)+2).
a(A004023(n)) = 2. - Michel Marcus, Sep 09 2015
a(A046413(n)) = 4. - Bruno Berselli, Sep 09 2015

Terms to a(280) in b-file from Hans Havermann, Aug 20 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) ib b-file from Max Alekseyev, May 04 2022

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

13, 156, 1520, 15912, 148512, 2042880, 14508000, 162493344, 1534205464, 16203253248, 144451398000, 2063316971520, 14903272088640, 158269280832000, 1614847741624320, 17205180696931968, 144444514193267496

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (terms 1..322 from Ray Chandler)
C. Caldwell, Sigma function.

Cf. A000203, A001270, A002283, A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102347, A102380, A112505, A147556, A295503, A366669.

DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)

a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A000203(A002283(n)). - Ray Chandler, Apr 22 2017

A095413 Total number of decimal digits of all distinct prime factors of the n-th repunit.

Original entry on oeis.org

0, 2, 3, 5, 5, 8, 7, 10, 9, 11, 11, 15, 13, 15, 17, 19, 17, 21, 19, 23, 24, 23, 23, 28, 27, 28, 27, 32, 30, 36, 31, 37, 35, 37, 38, 40, 38, 39, 40, 45, 42, 48, 45, 48, 48, 49, 47, 53, 50, 54, 54, 56, 55, 58, 58, 62, 60, 61, 59, 69, 63, 63, 69, 70, 67, 71, 67

Offset: 1

Labos Elemer, Jun 22 2004

			n=10: 10th repunit = 1111111111 = 11*41*271*9091; distinct prime factors have a total of 11 decimal digits, so a(10)=11.
n=27: 27th repunit = 111111111111111111111111111 = 3^3*37*757*333667*440334654777631, with 28 prime factor digits, a(27)=28.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A095414, A095415, A095416, A095417, A095418, A002275, A095407, A095370.

a[1] = 0; a[n_] := Total[IntegerLength /@ First /@ FactorInteger[(10^n - 1) /9]]; Array[a, 70] (* Giovanni Resta, Jul 09 2018 *)

a(n) = vecsum(apply(x->#Str(x), factor((10^n-1)/9)[,1])); \\ Michel Marcus, Jul 09 2018

a(n) = A095407(A002275(n)).

a(n) < A095370(n) + n. - Chai Wah Wu, Nov 04 2019

A095418 Excess of sum of all decimal digits of distinct prime factors for n-th repunit over corresponding digit-sum for repunit itself (which is n).

Original entry on oeis.org

-1, 0, 10, 0, 10, 20, 30, 17, 32, 26, 34, 35, 49, 53, 42, 51, 43, 74, 0, 56, 95, 77, 0, 81, 38, 94, 97, 106, 104, 80, 109, 123, 108, 96, 97, 132, 100, 65, 145, 136, 141, 184, 145, 173, 123, 99, 139, 172, 196, 120, 170, 176, 179, 213, 161, 169, 122, 201, 217, 184, 211, 216

Offset: 1

Labos Elemer, Jun 22 2004

			n=60: concatenated distinct-prime factor-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its digit sum is 244, so a(60) = 244 - 60 = 184.
The value of this excess-sum is zero if n=2,4,19,23.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A095402, A002275, A095370, A095413, A095414, A095415, A095416, A095417.

a[1] = -1; a[n_] := Total@ Flatten[IntegerDigits /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Array[a, 62] (* Giovanni Resta, Jul 19 2018 *)

a(n) = A095402(A002275(n)) - n = A095417(n) - n.

Data corrected by Giovanni Resta, Jul 19 2018

A095414 Excess of total number of distinct prime factor digits of n-th repunit over n, the number of digits of n-th repunit itself.

Original entry on oeis.org

-1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 2, 3, 0, 3, 0, 3, 3, 1, 0, 4, 2, 2, 0, 4, 1, 6, 0, 5, 2, 3, 3, 4, 1, 1, 1, 5, 1, 6, 2, 4, 3, 3, 0, 5, 1, 4, 3, 4, 2, 4, 3, 6, 3, 3, 0, 9, 2, 1, 6, 6, 2, 5, 0, 6, 3, 5, 0, 6, 1, 3, 6, 3, 3, 5, 2, 7, 2, 3, 0, 10, 2, 4

Offset: 1

Labos Elemer, Jun 22 2004

a(n) <= A095370(n) - 1 since the product of a k digit number and an m digit number has at least k+m-1 digits. - Chai Wah Wu, Nov 03 2019

			n=9: r9 = 111111111 = 3*3*37*333667, with a total of 9 digits among the distinct prime factors; the excess is a(9) = 9 - 9 = 0;
n=30: r30 = 111....1111 = 3*7*11*13*31*37*41*211*241*271*2161*9091*2906161, with a total of 36 digits among the distinct prime factors, so the excess a(30) = 36 - 30 = 6.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A002275, A095370, A095407, A095413, A095415, A095416, A095417, A095418.

a[1] = -1; a[n_] := Total[IntegerLength /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Array[a, 60] (* Giovanni Resta, Jul 16 2018 *)

a(n) = A095407(A002275(n)) - n = A095413(n) - n.

Data corrected by Giovanni Resta, Jul 16 2018

cp's OEIS Frontend

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

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A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

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A046053 Total number of prime factors of the repunit R(n) = (10^n-1)/9.

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A102347 Number of distinct prime factors of 10^n - 1.

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A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

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A070529 Number of divisors of repunit 111...111 (with n digits).

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