A185127 -id:A185127 - cp's OEIS frontend

A086578 a(n) = 7*(10^n - 1).

0, 63, 693, 6993, 69993, 699993, 6999993, 69999993, 699999993, 6999999993, 69999999993, 699999999993, 6999999999993, 69999999999993, 699999999999993, 6999999999999993, 69999999999999993, 699999999999999993, 6999999999999999993, 69999999999999999993, 699999999999999999993

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+7) = 7.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)).
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.
Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), this sequence (m=7), A100412 (m=8).

[7*(10^n -1): n in [0..20]]; // G. C. Greubel, Apr 14 2023

LinearRecurrence[{11,-10}, {0,63}, 31] (* G. C. Greubel, Apr 14 2023 *)

[7*(10^n -1) for n in range(21)] # G. C. Greubel, Apr 14 2023

a(n) = 7*9*A002275(n) = 7*A002283(n).
R(a(n)) = A086575(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 63*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 7*(exp(10*x) - exp(x)). - G. C. Greubel, Apr 14 2023

Edited by Jinyuan Wang, Aug 04 2021

A100412 a(n) = 8*10^n - 7.

Original entry on oeis.org

1, 73, 793, 7993, 79993, 799993, 7999993, 79999993, 799999993, 7999999993, 79999999993, 799999999993, 7999999999993, 79999999999993, 799999999999993, 7999999999999993, 79999999999999993, 799999999999999993

Offset: 0

Farideh Firoozbakht, Dec 08 2004

Also: Numbers n such that n is reversal(n)-th odd number. (This was the original definition. - Ed.)

All semiprimes in this sequence (n = 2, 4, 7, 9, 11, 16, 18, 23, 31, 32, 40, ...) are in A136543. - M. F. Hasler, Nov 03 2012

			793 is in the sequence because 793 is 397th odd number.
1 is in the sequence because 1 is the 1st odd number. - _M. F. Hasler_, Nov 03 2012

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A100413, A136543.

Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), A086578 (m=7), this sequence (m=8).

[8*10^n -7: n in [0..20]]; // G. C. Greubel, Apr 14 2023

Mathematica
```
Table[8*10^n-7, {n,0,20}]
```

A100412(n):=8*10^n-7$
makelist(A100412(n),n,0,17); /* Martin Ettl, Nov 08 2012 */

Vec((1+62*x)/((1-x)*(1-10*x)) + O(x^100)) \\ Colin Barker, Oct 14 2014

[8*10^n -7 for n in range(21)] # G. C. Greubel, Apr 14 2023

From Colin Barker, Oct 14 2014: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3).
G.f.: (1+62*x)/((1-x)*(1-10*x)). (End)
E.g.f.: 8*exp(10*x) - 7*exp(x). - G. C. Greubel, Apr 14 2023

Edited and extended to offset 0 by M. F. Hasler, Nov 03 2012

A117978 Triangular numbers with only even digits.

Original entry on oeis.org

0, 6, 28, 66, 406, 666, 820, 2080, 2628, 8646, 28680, 42486, 48828, 64620, 66066, 80200, 84666, 200028, 204480, 228826, 264628, 288420, 426426, 446040, 468028, 484620, 600060, 626640, 644680, 686206, 828828, 886446, 2222886, 2248260, 2862028, 2888406

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 03 2006

There are infinitely many terms. For example all 88...88 6 44...44 6 with an equal number of 8's and 4's (their triangular indices being A185127). - Shyam Sunder Gupta, Aug 16 2025

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.

Intersection of A000217 and A014263.

Cf. A117960, A185127.

Select[Accumulate[Range[0,5000]],Count[IntegerDigits[#],?(OddQ)] ==0&] (* _Harvey P. Dale, Oct 21 2011 *)

Corrected (a(32) was in error) and extended by Harvey P. Dale, Oct 21 2011

A259050 Numbers k such that 3*R_k + 10^k - 2 is prime, where R_k = 11...11 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 2, 4, 6, 94, 160, 360, 1470, 2898, 3094, 3112, 15698, 17956, 42262, 111032, 249550

Offset: 1

Robert Price, Jun 17 2015

Also, numbers k such that (4*10^k - 7)/3 is prime.
Terms from Kamada data.
a(16) > 2*10^5.
The corresponding primes are a subset of the palindromes A185127 with a(n)+1 digits [1, 3 repeated a(n)-1 times, 1]: 11, 131, 13331, 1333331, ..., which can be expressed as 2*6-1, 2*66-1, 2*6666-1, 2*666666-1, ... . - Hugo Pfoertner, Jul 22 2020

			For k=4, 3*R_4 + 10^k - 2 = 3333 + 10000 - 2 = 13331 which is prime.

Makoto Kamada, Near-repdigit numbers of the form ABB...BBA.
Makoto Kamada, Prime numbers of the form 133...331.
Index entries for primes involving repunits.

Cf. A002275, A069882, A185127.

[n: n in [0..500] | IsPrime((4*10^n-7) div 3)]; // Vincenzo Librandi, Jun 18 2015

Select[Range[0, 200000], PrimeQ[(4*10^#-7)/3] &]

for(k=1,1500,if(ispseudoprime(4*(10^k-1)/3-1),print1(k,", "))) \\ Hugo Pfoertner, Jul 22 2020

a(16) from Kamada data by Tyler Busby, May 03 2024

A069882 Numbers n such that n and 2n-1 are both palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 66, 666, 6666, 66666, 666666, 6666666, 66666666, 666666666, 6666666666, 66666666666, 666666666666, 6666666666666, 66666666666666, 666666666666666, 6666666666666666, 66666666666666666, 666666666666666666

Offset: 1

Amarnath Murthy, Apr 30 2002

From Chai Wah Wu, Jul 20 2020: (Start)
Theorem: a(n) = 2*(10^(n-5)-1)/3 for n > 5.
Proof: clearly 2*(10^m-1)/3 are terms of this sequence. Next we show that all terms > 10 are of the form 2*(10^m-1)/3. Let k > 10 be a term of the sequence. Let x be the first digit (and thus also the last digit) of k. If x <> 6 then it is easy to show that the first and last digit of 2k-1 will not be the same. Thus x = 6. Let the digits of k be written as 6y****y6. Similarly if y <> 6 then again the second digit of 2k-1 will not be the same as the second to last digit of 2k-1. Continuing in this manner, this shows that k written in decimal is a sequence of 6's.
(End)

			66 is a member as 2*66 - 1 = 131 is also a palindrome.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002280, A069881, A185127, A259050.

From Chai Wah Wu, Jul 20 2020: (Start)
a(n) = 2*(10^(n-5)-1)/3 for n > 5.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 7.
G.f.: x*(50*x^6 - 9*x^5 - 9*x^4 - 9*x^3 - 9*x^2 - 9*x + 1)/((x - 1)*(10*x - 1)).
(End)

More terms from Hans Havermann, Jul 06 2002

cp's OEIS Frontend

A086578 a(n) = 7*(10^n - 1).

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A100412 a(n) = 8*10^n - 7.

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A117978 Triangular numbers with only even digits.

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A259050 Numbers k such that 3*R_k + 10^k - 2 is prime, where R_k = 11...11 is the repunit (A002275) of length k.

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Programs

Magma

Mathematica

PARI

Extensions

A069882 Numbers n such that n and 2n-1 are both palindromes.

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Keywords

Comments

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Crossrefs

Formula

Extensions