A028988 -id:A028988

A096503

Euler-phi of these numbers is a decimal repdigit.

+10
17

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 23, 24, 30, 46, 67, 69, 89, 92, 115, 134, 138, 178, 184, 223, 230, 276, 446, 669, 892, 1043, 1115, 1338, 1341, 1784, 2086, 2230, 2676, 2682, 446669, 666667, 893338, 895043, 902423, 1333334, 1340007, 1786676

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

T. D. Noe, Table of n, a(n) for n = 1..182

D. Bressoud, CNT.m Computational Number Theory Mathematica package.

EXAMPLE

n=88888892, A000010(n)=44444444.

Regular solutions: if x=repdigit+1 is prime, then phi[x]=repdigit (see A028988).

MATHEMATICA

Needs["CNT`"]; t = {PhiInverse[1]}; Do[n = FromDigits[Table[i, {j}]]; AppendTo[t, PhiInverse[n]], {j, 18}, {i, 2, 8, 2}]; t2 = Union[Flatten[t]]; t (* T. D. Noe, Feb 25 2014 *)

Select[Range[2*10^5], Length@ Union@ IntegerDigits@ EulerPhi@ # == 1 &] (* Michael De Vlieger, Jul 02 2016 *)

PROG

(PARI) isok(n) = d = digits(eulerphi(n)); vecmin(d) == vecmax(d); \\ Michel Marcus, Feb 25 2014

CROSSREFS

Cf. A010785, A028988.

KEYWORD

base,nonn

AUTHOR

Labos Elemer, Jul 12 2004

STATUS

approved

A048328

Numbers that are repdigits in base 3.

+10
10

0, 1, 2, 4, 8, 13, 26, 40, 80, 121, 242, 364, 728, 1093, 2186, 3280, 6560, 9841, 19682, 29524, 59048, 88573, 177146, 265720, 531440, 797161, 1594322, 2391484, 4782968, 7174453, 14348906, 21523360, 43046720, 64570081, 129140162, 193710244, 387420488, 581130733

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Case for base 2 see A000225: 2^n - 1.

If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Repdigit.

Index entries for 3-automatic sequences.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

FORMULA

G.f.: (2*x^2+x)/(1-4*x^2+3*x^4). - Alois P. Heinz, Sep 23 2012

Sum_{n>=1} 1/a(n) = 3 * A214369 = 2.04646050781571420028... - Amiram Eldar, Jan 21 2022

a(n) = (3^(n/2)*(sqrt(3) + 2 - (-1)^n*(sqrt(3) - 2)) - 3 - (-1)^n)/4. - Stefano Spezia, Feb 18 2022

MAPLE

nmax := 35; a(0) := 0: for n from 1 to nmax do a(2*n) := a(2*n-2) + 2*3^(n-1); od: a(1) := 1: for n from 1 to nmax do a(2*n+1) := 1*a(2*n-1) + 3^n; od: seq(a(n), n=0..nmax);

# End program 1

with(GraphTheory): G := PathGraph(5): A:= AdjacencyMatrix(G): nmax := nmax; for n from 1 to nmax+1 do B(n) := A^n; b(n) := add(B(n)[1, k], k=1..4); a1(n-1) := b(n)-1; od: seq(a1(n), n=0..nmax);

# End program 2

# From Johannes W. Meijer, May 29 2010, revised Sep 23 2012

# third Maple program:

a:= n->(<<0|1>, <-3|4>>^iquo(n, 2, 'r').`if`(r=0, <<0, 2>>, <<1, 4>>))[1, 1]:

seq (a(n), n=0..60); # Alois P. Heinz, Sep 23 2012

MATHEMATICA

Rest[FromDigits[#, 3]&/@Flatten[Table[{PadRight[{1}, n, 1], PadRight[{2}, n, 2]}, {n, 0, 20}], 1]] (* Harvey P. Dale, Feb 03 2011 *)

PROG

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -3, 0, 4, 0]^n*[0; 1; 2; 4])[1, 1] \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Bisections: A024023 and A003462.

Cf. A000225, A010785, A028987, A028988, A033016, A038754, A068911, A124302, A214369.

KEYWORD

nonn,base,easy

AUTHOR

Patrick De Geest, Feb 15 1999

STATUS

approved

A054268

Sum of composite numbers between prime p and nextprime(p) is a repdigit.

+10
10

3, 5, 109, 111111109, 259259257

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

No additional terms below 472882027.

No additional terms below 10^58. - Chai Wah Wu, Jun 01 2024

LINKS

Table of n, a(n) for n=1..5.

Eric Weisstein's World of Mathematics, Repdigit

FORMULA

Numbers A000040(n) for n > 1 such that A001043(n)*(A001223(n)-1)/2 is in A010785. - Chai Wah Wu, Aug 12 2014

EXAMPLE

a(5) is ok since between 259259257 and nextprime 259259261 we get the sum 259259258 + 259259259 + 259259260 which yield repdigit 777777777.

MATHEMATICA

repQ[n_]:=Count[DigitCount[n], 0]==9; Select[Prime[Range[2, 14500000]], repQ[Total[Range[#+1, NextPrime[#]-1]]]&] (* Harvey P. Dale, Jan 29 2011 *)

PROG

(Python)

from sympy import prime

A054268 = [prime(n) for n in range(2, 10**5) if len(set(str(int((prime(n+1)-prime(n)-1)*(prime(n+1)+prime(n))/2)))) == 1]

# Chai Wah Wu, Aug 12 2014

(Python)

from itertools import count, islice

from sympy import isprime, nextprime

from sympy.abc import x, y

from sympy.solvers.diophantine.diophantine import diop_quadratic

def A054268_gen(): # generator of terms

for l in count(1):

c = []

for m in range(1, 10):

k = m*(10**l-1)//9<<1

for a, b in diop_quadratic((x-y-1)*(x+y)-k):

if isprime(b) and a == nextprime(b):

c.append(b)

yield from sorted(c)

A054268_list = list(islice(A054268_gen(), 5)) # Chai Wah Wu, Jun 01 2024

CROSSREFS

Cf. A010785, A028987, A028988, A046933, A054264, A054265, A054266, A054267.

KEYWORD

nonn,base,hard

AUTHOR

Patrick De Geest, Apr 15 2000

EXTENSIONS

Offset changed by Andrew Howroyd, Aug 14 2024

STATUS

approved

A048329

Numbers that are repdigits in base 4.

+10
5

0, 1, 2, 3, 5, 10, 15, 21, 42, 63, 85, 170, 255, 341, 682, 1023, 1365, 2730, 4095, 5461, 10922, 16383, 21845, 43690, 65535, 87381, 174762, 262143, 349525, 699050, 1048575, 1398101, 2796202, 4194303, 5592405, 11184810, 16777215, 22369621, 44739242, 67108863

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Eric Weisstein's World of Mathematics, Repdigit.

Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-4).

FORMULA

G.f.: x*(1+2*x+3*x^2) / ( (x-1)*(4*x^3-1)*(1+x+x^2) ) with a(n) = 5*a(n-3) - 4*a(n-6). - R. J. Mathar, Mar 15 2015

Sum_{n>=1} 1/a(n) = (11/2) * A248721 = 2.31603727318383077512... - Amiram Eldar, Jan 21 2022

EXAMPLE

10_10 = 22_4, 15_10 = 33_4, 5461_10 = 1111111_4.

MAPLE

a:= n-> (1+irem(n+2, 3))*(4^iquo(n+2, 3)-1)/3:

seq(a(n), n = 0..45);

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 4], {n, 0, 40}, {d, 3}]]](* Vincenzo Librandi, Feb 06 2014 *)

LinearRecurrence[{0, 0, 5, 0, 0, -4}, {0, 1, 2, 3, 5, 10}, 40] (* Harvey P. Dale, Jul 11 2023 *)

PROG

(Magma) [0] cat [k:k in [1..10^7]| #Set(Intseq(k, 4)) eq 1]; // Marius A. Burtea, Oct 11 2019

CROSSREFS

Cf. A010785, A033017, A028987, A028988, A248721.

Base 4 repdigits 1,2,3 (trisections): A002450, A020988, A024036.

KEYWORD

nonn,base,easy

AUTHOR

Patrick De Geest, Feb 15 1999

STATUS

approved

A048333

Numbers that are repdigits in base 8.

+10
5

0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 73, 146, 219, 292, 365, 438, 511, 585, 1170, 1755, 2340, 2925, 3510, 4095, 4681, 9362, 14043, 18724, 23405, 28086, 32767, 37449, 74898, 112347, 149796, 187245, 224694, 262143, 299593, 599186, 898779

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

For the general case, the sequence of numbers that are repdigits in base b > 1 satisfies the recurrence a(n) = (b+1)*a(n-b+1) - b*a(n-2*(b-1)) for n >= 2(b-1) with g.f.: (sum_{1 <= i < b} i*x^i)/(1 - (b+1)*x^(b-1) + bx^(2(b-1))). - Chai Wah Wu, May 30 2016

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Eric Weisstein's World of Mathematics, Repdigit.

Index entries for 2-automatic sequences.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,9,0,0,0,0,0,0,-8).

FORMULA

From Chai Wah Wu, May 30 2016: (Start)

a(n) = 9*a(n-7) - 8*a(n-14) for n > 13.

G.f.: x*(7*x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(8*x^14 - 9*x^7 + 1). (End)

Sum_{n>=1} 1/a(n) = (363/20) * A248725 = 2.92153624531838250201... - Amiram Eldar, Jan 21 2022

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 8], {n, 0, 40}, {d, 7}]]] (* Vincenzo Librandi, Feb 06 2014 *)

LinearRecurrence[{0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, -8}, {0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54}, 50] (* Harvey P. Dale, Dec 09 2018 *)

PROG

(PARI) is(n)=#Set(digits(n, 8))==1 \\ Charles R Greathouse IV, Feb 15 2017

CROSSREFS

Cf. A010785, A033021, A028987, A028988, A248725.

KEYWORD

nonn,base,easy

AUTHOR

Patrick De Geest, Feb 15 1999

EXTENSIONS

Changed offset from 1 to 0 by Vincenzo Librandi, Feb 06 2014

STATUS

approved

A048330

Numbers that are repdigits in base 5.

+10
4

0, 1, 2, 3, 4, 6, 12, 18, 24, 31, 62, 93, 124, 156, 312, 468, 624, 781, 1562, 2343, 3124, 3906, 7812, 11718, 15624, 19531, 39062, 58593, 78124, 97656, 195312, 292968, 390624, 488281, 976562, 1464843, 1953124, 2441406, 4882812, 7324218, 9765624

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Eric Weisstein's World of Mathematics, Repdigit.

FORMULA

Conjecture: G.f.: x*(1+2*x+3*x^2+4*x^3) / ( (x-1)*(1+x)*(x^2+1)*(5*x^4-1) ) with a(n) = 6*a(n-4) - 5*a(n-8). - R. J. Mathar, Mar 15 2015

Sum_{n>=1} 1/a(n) = (25/3) * A248722 = 2.51444877998310381623... - Amiram Eldar, Jan 21 2022

EXAMPLE

12_10 = 22_5, 18_10 = 33_5, 7812_10 = 222222_5.

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 5], {n, 0, 40}, {d, 4}]]] (* Vincenzo Librandi, Feb 06 2014 *)

PROG

(Magma) [0] cat [k:k in [1..10^7]| #Set(Intseq(k, 5)) eq 1]; // Marius A. Burtea, Oct 11 2019

CROSSREFS

Cf. A010785, A033018, A028987, A028988, A248722.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest, Feb 15 1999

EXTENSIONS

Offset changed from 1 to 0 by Vincenzo Librandi, Feb 06 2014

STATUS

approved

A048331

Numbers that are repdigits in base 6.

+10
4

0, 1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 43, 86, 129, 172, 215, 259, 518, 777, 1036, 1295, 1555, 3110, 4665, 6220, 7775, 9331, 18662, 27993, 37324, 46655, 55987, 111974, 167961, 223948, 279935, 335923, 671846, 1007769, 1343692, 1679615, 2015539

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Repdigit.

FORMULA

Conjecture: G.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4) / ( (x-1)*(x^4+x^3+x^2+x+1)*(6*x^5-1) ) with a(n) = 7*a(n-5) - 6*a(n-10). - R. J. Mathar, Mar 15 2015

Sum_{n>=1} 1/a(n) = (137/12) * A248723 = 2.67320256903907177403... - Amiram Eldar, Jan 21 2022

EXAMPLE

14_10 = 22_6, 21_10_ = 33_6, 9331_10_ = 111111_6.

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 6], {n, 0, 40}, {d, 5}]]] (* Vincenzo Librandi, Feb 06 2014 *)

PROG

(Magma) [0] cat [k:k in [1..2*10^6]| #Set(Intseq(k, 6)) eq 1]; // Marius A. Burtea, Oct 11 2019

CROSSREFS

Cf. A010785, A033019, A028987, A028988, A248723.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest, Feb 15 1999

STATUS

approved

A048332

Numbers that are repdigits in base 7.

+10
4

0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 57, 114, 171, 228, 285, 342, 400, 800, 1200, 1600, 2000, 2400, 2801, 5602, 8403, 11204, 14005, 16806, 19608, 39216, 58824, 78432, 98040, 117648, 137257, 274514, 411771, 549028, 686285, 823542, 960800

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Eric Weisstein's World of Mathematics, Repdigit.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,8,0,0,0,0,0,-7).

FORMULA

From Chai Wah Wu, May 30 2016: (Start)

a(n) = 8*a(n-6) - 7*a(n-12) for n > 11.

G.f.: x*(6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(7*x^12 - 8*x^6 + 1). (End)

a(n) = (n - 6*floor((n-1)/6))*(7^floor((n+5)/6) - 1)/6. - Ilya Gutkovskiy, May 30 2016

Sum_{n>=1} 1/a(n) = (147/10) * A248724 = 2.80637791743084519957... - Amiram Eldar, Jan 21 2022

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 7], {n, 0, 40}, {d, 6}]]] (* Vincenzo Librandi, Feb 06 2014 *)

LinearRecurrence[{0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -7}, {0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40}, 25] (* G. C. Greubel, May 30 2016 *)

PROG

(Python)

A048332_list = [0] + [int(d*l, 7) for l in range(1, 10) for d in '123456'] # Chai Wah Wu, May 30 2016

CROSSREFS

Cf. A010785, A033020, A028987, A028988, A248724.

KEYWORD

nonn,base,easy

AUTHOR

Patrick De Geest, Feb 15 1999

STATUS

approved

A048334

Numbers that are repdigits in base 9.

+10
3

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 91, 182, 273, 364, 455, 546, 637, 728, 820, 1640, 2460, 3280, 4100, 4920, 5740, 6560, 7381, 14762, 22143, 29524, 36905, 44286, 51667, 59048, 66430, 132860, 199290, 265720, 332150, 398580

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Eric Weisstein's World of Mathematics, Repdigit.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,-9).

FORMULA

G.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7) / ( (x-1) *(1+x) *(x^2+1) *(3*x^4-1) *(3*x^4+1) *(x^4+1) ). - R. J. Mathar, Mar 14 2015

a(n) = 10*a(n-8) -9*a(n-16). - R. J. Mathar, Mar 14 2015

Sum_{n>=1} 1/a(n) = (761/35) * A248726 = 3.02323812974071904119... - Amiram Eldar, Jan 21 2022

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 9], {n, 0, 40}, {d, 8}]]] (* Vincenzo Librandi, Feb 06 2014 *)

Table[FromDigits[IntegerDigits[(n-9*Floor[(n-1)/9])*(10^Floor[(n+8)/9]-1)/9], 9], {n, 0, 50}] (* Zak Seidov, Mar 15 2015 *)

f[n_] := Block[{r = FromDigits[#, 9] & /@ (Table[1, {#}] & /@ Range@ n)},

Sort@ Flatten[Times[r, #] & /@ Range@ 8]]; f[6] (* Michael De Vlieger, Mar 15 2015 *)

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, -9}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70}, 47] (* Ray Chandler, Jul 15 2015 *)

PROG

(PARI) lista(nn) = for (n=0, nn, if ((n==0) || (#Set(digits(n, 9)) == 1), print1(n, ", "))); \\ Michel Marcus, Mar 17 2015

CROSSREFS

Cf. A010785, A033022, A028987, A028988, A248726.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest, Feb 15 1999

STATUS

approved

A048340

a(n) in base 16 is a repdigit.

+10
3

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 273, 546, 819, 1092, 1365, 1638, 1911, 2184, 2457, 2730, 3003, 3276, 3549, 3822, 4095, 4369, 8738, 13107, 17476, 21845, 26214, 30583

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Eric Weisstein's World of Mathematics, Repdigit

FORMULA

From Chai Wah Wu, May 30 2016: (Start)

a(n) = 17*a(n-15) - 16*a(n-30) for n > 29.

x*(15*x^14 + 14*x^13 + 13*x^12 + 12*x^11 + 11*x^10 + 10*x^9 + 9*x^8 + 8*x^7 + 7*x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(16*x^30 - 17*x^15 + 1).

(End)

a(n) = (n - 15*floor((n-1)/15))*(16^floor((n+14)/15) - 1)/15. - Ilya Gutkovskiy, May 30 2016

MATHEMATICA

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 16], {n, 0, 50}, {d, 15}]]] (* Vincenzo Librandi, Feb 06 2014 *)

PROG

(Python)

A048340_list = [0] + [int(d*l, 16) for l in range(1, 10) for d in '123456789abcdef'] # Chai Wah Wu, May 30 2016

CROSSREFS

Cf. A010785, A033029, A028987, A028988.

KEYWORD

nonn,base,easy

AUTHOR

Patrick De Geest, Feb 15 1999

STATUS

approved