A002281 -id:A002281 - cp's OEIS frontend

A332171 a(n) = 7(10^(2n+1)-1)/9 - 610^n.

1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777

Offset: 0

M. F. Hasler, Feb 06 2020

For n == 0 or n == 2 (mod 6), there is no obvious divisibility pattern.

According to M. Kamada, n = 116 is the only index of a prime up to n = 10^5.

Makoto Kamada, Factorization of 77...77177...77.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332170 .. A332179 (variants with different middle digit 2, ..., 9).

Array[7 (10^(2 # + 1) - 1)/9 - 6*10^# &, 15, 0] (* or *)
CoefficientList[Series[(1 + 606 x - 1300 x^2)/((1 - x) (1 - 10 x) (1 - 100 x)), {x, 0, 15}], x] (* Michael De Vlieger, Feb 08 2020 *)
Table[FromDigits[Join[PadRight[{},n,7],{1},PadRight[{},n,7]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{1,717,77177},20] (* Harvey P. Dale, Apr 04 2024 *)

apply( {A332171(n)=10^(n*2+1)\9*7-6*10^n}, [0..15])

Vec((1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^15)) \\ Colin Barker, Feb 07 2020

def A332171(n): return 10**(n*2+1)//9*7-6*10^n

a(n) = 7*A138148(n) + 10^n.
For n == 1 (mod 3), 3 | a(n) and a(n)/3 = 259*(10^(2n+1)-1)/999 - 2*10^n;
for n == 3 or 5 (mod 6), 13 | a(n) and a(n)/13 = (A(n)-1)*10^n + B(n), where A(n) (resp. B(n)) are the n leftmost (resp. rightmost) digits of 59829*(10^(ceiling(n/6)*6)-1)/(10^6-1).
From Colin Barker, Feb 07 2020: (Start)
G.f.: (1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.
(End)
E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 54*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020

A332179 a(n) = 7(10^(2n+1)-1)/9 + 210^n.

Original entry on oeis.org

9, 797, 77977, 7779777, 777797777, 77777977777, 7777779777777, 777777797777777, 77777777977777777, 7777777779777777777, 777777777797777777777, 77777777777977777777777, 7777777777779777777777777, 777777777777797777777777777, 77777777777777977777777777777, 7777777777777779777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183183 = {1, 2, 8, 19, 20, 212, 280, ...} for the indices of primes.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077796-1)/2 = A183183: indices of primes.
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332178 (variants with different middle digit 1, ..., 8).

A332179 := n -> 7*(10^(n*2+1)-1)/9 + 2*10^n;

Array[7 (10^(2 # + 1) - 1)/9 + 2*10^# &, 15, 0]

apply( {A332179(n)=10^(n*2+1)\9*7+2*10^n}, [0..15])

def A332179(n): return 10**(n*2+1)//9*7+2*10^n

a(n) = 7*A138148(n) + 9*10^n.
G.f.: (9 - 202*x - 500*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332178 a(n) = 7*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

8, 787, 77877, 7778777, 777787777, 77777877777, 7777778777777, 777777787777777, 77777777877777777, 7777777778777777777, 777777777787777777777, 77777777777877777777777, 7777777777778777777777777, 777777777777787777777777777, 77777777777777877777777777777, 7777777777777778777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183182 = {1, 3, 39, 54, 168, 240, ...} for the indices of primes.

Makoto Kamada, Factorization of 77...77877...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077793-1)/2 = A183182: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

A332178 := n -> 7*(10^(n*2+1)-1)/9 + 10^n;

Array[7 (10^(2 # + 1) - 1)/9 + 10^# &, 15, 0]

apply( {A332178(n)=10^(n*2+1)\9*7+10^n}, [0..15])

def A332178(n): return 10**(n*2+1)//9*7+10^n

a(n) = 7*A138148(n) + 8*10^n.
G.f.: (8 - 101*x - 600*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A180160 (sum of digits) mod (number of digits) of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2

Offset: 0

Reinhard Zumkeller, Aug 15 2010

a(n) = A007953(n) mod A055642(n);
a(A061383(n)) = 0; a(A180157(n)) > 0;
a(repdigits)=0: a(A010785(n))=0: a(A002275(n))=0: a(A002276(n))=0: a(A002277(n))=0: a(A002278(n))=0: a(4(n))=0: a(A002279(n))=0: a(A002280(n))=0: a(A002281(n))=0: a(A002282(n))=0: a(A002283(n))=0;
A123522 gives smallest m such that a(m) = n.

Reinhard Zumkeller, Table of n, a(n) for n = 0..25000

Cf. A002275-A002283, A007953, A010785, A055642, A061383, A123522, A180157, A374097.

A180160[n_] := If[n == 0, 0, Mod[Total[#], Length[#]] & [IntegerDigits[n]]];
Array[A180160, 100, 0] (* Paolo Xausa, Jun 30 2024 *)
Join[{0},Table[Mod[Total[IntegerDigits[n]],IntegerLength[n]],{n,110}]] (* Harvey P. Dale, Jul 30 2025 *)

A178769 a(n) = (5*10^n + 13)/9.

Original entry on oeis.org

2, 7, 57, 557, 5557, 55557, 555557, 5555557, 55555557, 555555557, 5555555557, 55555555557, 555555555557, 5555555555557, 55555555555557, 555555555555557, 5555555555555557, 55555555555555557, 555555555555555557, 5555555555555555557, 55555555555555555557, 555555555555555555557

Offset: 0

Bruno Berselli, Jun 13 2010

Bruno Berselli, Table of n, a(n) for n = 0..1000.
Bruno Berselli, A property that includes the numbers of the form 5..57.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A165246 (..17, 117, 1117,..), A173193 (..27, 227, 2227,..), A173766 (..37, 337, 3337,..), A173772 (..47, 447, 4447,..), A067275 (..67, 667, 6667,..), A002281 (..77, 777, 7777,..), A173812 (..87, 887, 8887,..), A173833 (..97, 997, 9997,..).

Cf. A093143.

List([0..20], n -> (5*10^n+13)/9); # G. C. Greubel, Jan 24 2019

[(5*10^n+13)/9: n in [0..20]]; // Vincenzo Librandi, Jun 06 2013

CoefficientList[Series[(2 - 15 x) / ((1 - x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{11,-10},{2,7},20] (* Harvey P. Dale, Feb 28 2017 *)

vector(20, n, n--; (5*10^n+13)/9) \\ G. C. Greubel, Jan 24 2019

[(5*10^n+13)/9 for n in (0..20)] # G. C. Greubel, Jan 24 2019

a(n)^(4*k+2) + 1 == 0 (mod 250) for n > 1, k >= 0.
G.f.: (2-15*x)/((1-x)*(1-10*x)).
a(n) - 11*a(n-1) + 10*a(n-2) = 0 (n > 1).
a(n) = a(n-1) + 5*10^(n-1) = 10*a(n-1) - 13 for n > 0.
a(n) = 1 + Sum_{i=0..n} A093143(i). - Bruno Berselli, Feb 16 2015
E.g.f.: exp(x)*(5*exp(9*x) + 13)/9. - Elmo R. Oliveira, Sep 09 2024

A366596 Repdigit numbers that are divisible by 7.

Original entry on oeis.org

0, 7, 77, 777, 7777, 77777, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 7777777, 77777777, 777777777, 7777777777, 77777777777, 111111111111, 222222222222, 333333333333, 444444444444, 555555555555, 666666666666, 777777777777

Offset: 1

Kritsada Moomuang, Oct 14 2023

7 divides a repdigit iff it consists of only digit 7, or has length 6*k (for any digit).

Repdigit remainders A010785(k) mod 7 have period 54. - Karl-Heinz Hofmann, Dec 04 2023

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..2329
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1000001,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000000).

Intersection of A008589 and A010785.
Cf. A002281 (a subsequence).
Cf. A305322 (divisor 3), A002279 (divisor 5), A083118 (the impossible divisors).

r(n) = 10^((n+8)\9)\9*((n-1)%9+1); \\ A010785
lista(nn) = select(x->!(x%7), vector(nn, k, r(k-1))); \\ Michel Marcus, Oct 26 2023

def A366596(n):
    digitlen, digit = (n+12)//14*6, (n+12)%14-4
    if digit < 1: digitlen += digit - 1; digit = 7
    return 10**digitlen // 9 * digit # Karl-Heinz Hofmann, Dec 04 2023

From Karl-Heinz Hofmann, Dec 04 2023: (Start)
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14) + 41),   for (n-2) mod 14 >  4.
a(n) = (10^(6*floor((n-2)/14) + 6)-1)/9*(((n-2) mod 14)-4), for (n-2) mod 14 >  4.
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14)*9 + 7),  for (n-2) mod 14 <= 4.
a(n) = (10^(6*floor((n-2)/14) + 1 + ((n-2) mod 14))-1)/9*7, for (n-2) mod 14 <= 4.
(End)

A332170 a(n) = 7(10^(2n+1)-1)/9 - 710^n.

Original entry on oeis.org

0, 707, 77077, 7770777, 777707777, 77777077777, 7777770777777, 777777707777777, 77777777077777777, 7777777770777777777, 777777777707777777777, 77777777777077777777777, 7777777777770777777777777, 777777777777707777777777777, 77777777777777077777777777777, 7777777777777770777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

A332170 := n -> 7*(10^(2*n+1)-1)/9-7*10^n;

Array[7 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]

apply( {A332170(n)=(10^(n*2+1)\9-10^n)*7}, [0..15])

def A332170(n): return (10**(n*2+1)//9-10^n)*7

a(n) = 7*A138148(n) = A002281(2n+1) - 7*A011557(n).
G.f.: 7*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 11, 2, 0, 0, 111, 22, 3, 0, 0, 1111, 222, 33, 4, 0, 0, 11111, 2222, 333, 44, 5, 0, 0, 111111, 22222, 3333, 444, 55, 6, 0, 0, 1111111, 222222, 33333, 4444, 555, 66, 7, 0, 0, 11111111, 2222222, 333333, 44444, 5555, 666, 77, 8, 0

Offset: 0

Stefano Spezia, Sep 14 2023

			The array begins:
  0,     0,     0,     0,     0,     0, ...
  0,     1,     2,     3,     4,     5, ...
  0,    11,    22,    33,    44,    55, ...
  0,   111,   222,   333,   444,   555, ...
  0,  1111,  2222,  3333,  4444,  5555, ...
  0, 11111, 22222, 33333, 44444, 55555, ...
  ...

Cf. A000004 (n=0 or k=0), A001477 (n=1), A002275 (k=1), A002276 (k=2), A002277 (k=3), A002278 (k=4), A002279 (k=5), A002280 (k=6), A002281 (k=7), A002282 (k=8), A002283 (k=9), A008593 (n=2), A053422 (main diagonal), A105279 (k=10), A132583, A177769 (n=3), A365645 (antidiagonal sums), A365646.

A[n_,k_]:=k(10^n-1)/9; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

O.g.f.: x*y/((1 - x)*(1 - 10*x)*(1 - y)^2).
E.g.f.: y*exp(x+y)*(exp(9*x) - 1)/9.
A(n, 11) = A132583(n-1) for n > 0.
A(n, 12) = A073551(n+1) for n > 0.

A285094 Corresponding values of geometric means of digits of numbers from A061430.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 0, 2, 4, 0, 3, 0, 2, 4, 6, 0, 5, 0, 6, 0, 7, 0, 4, 8, 0, 3, 6, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 3, 0, 2, 0, 0, 0, 0, 2, 4, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 4, 0

Offset: 0

Jaroslav Krizek, Apr 14 2017

Jaroslav Krizek, Table of n, a(n) for n = 0..3000

Cf. A061430 (numbers with integer geometric mean of digits in base 10).

Sequences of numbers n such that a(n) = k for k = 0 - 9: A011540 (k = 0), A002275 (k = 1), A061426 (k = 2), A061427 (k = 3), A061428 (k = 4), A002279 (k = 5), A061429 (k = 6), A002281 (k = 7), A002282 (k = 8), A002283 (k = 9).

[0] cat [Floor(&*Intseq(n) ^ (1/#Intseq(n))): n in [1..100000] | IsIntegral(&*Intseq(n) ^ (1/#Intseq(n)))];

A332172 a(n) = 7(10^(2n+1)-1)/9 - 510^n.

Original entry on oeis.org

2, 727, 77277, 7772777, 777727777, 77777277777, 7777772777777, 777777727777777, 77777777277777777, 7777777772777777777, 777777777727777777777, 77777777777277777777777, 7777777777772777777777777, 777777777777727777777777777, 77777777777777277777777777777, 7777777777777772777777777777777

Offset: 0

M. F. Hasler, Feb 06 2020

Indices of prime terms: {0, 1, 3, 7, 10, 12, 480, 949, ...} = A183178.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 (analog with middle digit 1).
Cf. (A077777-1)/2 = A183178: indices of primes.
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

A332172 := n -> 7*(10^(n*2+1)-1)/9 -5*10^n;

Array[7 (10^(2 # +1)-1)/9 -5*10^# &, 15, 0]

apply( {A332172(n)=10^(n*2+1)\9*7-5*10^n}, [0..25])

def A332172(n): return 10**(n*2+1)//9*7-5*10^n

a(n) = 7*A138148(n) + 2*10^n.
G.f.: (2 + 505*x - 1200*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.

cp's OEIS Frontend

A332171 a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.

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A332179 a(n) = 7*(10^(2n+1)-1)/9 + 2*10^n.

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A332178 a(n) = 7*(10^(2n+1)-1)/9 + 10^n.

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A180160 (sum of digits) mod (number of digits) of n in decimal representation.

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A178769 a(n) = (5*10^n + 13)/9.

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A366596 Repdigit numbers that are divisible by 7.

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A332170 a(n) = 7*(10^(2n+1)-1)/9 - 7*10^n.

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A332171 a(n) = 7(10^(2n+1)-1)/9 - 610^n.

A332179 a(n) = 7(10^(2n+1)-1)/9 + 210^n.

A332170 a(n) = 7(10^(2n+1)-1)/9 - 710^n.

A332172 a(n) = 7(10^(2n+1)-1)/9 - 510^n.