A028988 -id:A028988 - cp's OEIS frontend

A096503 Euler-phi of these numbers is a decimal repdigit.

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

Offset: 1

Labos Elemer, Jul 12 2004

			n=88888892, A000010(n)=44444444.
Regular solutions: if x=repdigit+1 is prime, then phi[x]=repdigit (see A028988).

T. D. Noe, Table of n, a(n) for n = 1..182
D. Bressoud, CNT.m Computational Number Theory Mathematica package.

Cf. A010785, A028988.

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 *)

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

A048328 Numbers that are repdigits in base 3.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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

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).

Bisections: A024023 and A003462.

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

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

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

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

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

A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit.

Original entry on oeis.org

3, 5, 109, 111111109, 259259257

Offset: 1

Patrick De Geest, Apr 15 2000

No additional terms below 472882027.

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

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

Eric Weisstein's World of Mathematics, Repdigit

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

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

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

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

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

Offset changed by Andrew Howroyd, Aug 14 2024

A048329 Numbers that are repdigits in base 4.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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

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).

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

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

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

a:= n-> (1+irem(n+2, 3))*(4^iquo(n+2,3)-1)/3:
seq(a(n), n = 0..45);

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 *)

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

A048333 Numbers that are repdigits in base 8.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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

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).

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

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 *)

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

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

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

A048330 Numbers that are repdigits in base 5.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Repdigit.

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

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

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

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

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

A048331 Numbers that are repdigits in base 6.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Repdigit.

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

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

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

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

A048332 Numbers that are repdigits in base 7.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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).

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

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 *)

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

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

A048334 Numbers that are repdigits in base 9.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

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).

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

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 *)

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

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

A048340 a(n) in base 16 is a repdigit.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Feb 15 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Eric Weisstein's World of Mathematics, Repdigit

Cf. A010785, A033029, A028987, A028988.

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

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

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

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A096503 Euler-phi of these numbers is a decimal repdigit.

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A048328 Numbers that are repdigits in base 3.

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A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit.

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A048329 Numbers that are repdigits in base 4.

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A048333 Numbers that are repdigits in base 8.

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A048330 Numbers that are repdigits in base 5.

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A048331 Numbers that are repdigits in base 6.

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