A014824 -id:A014824 - cp's OEIS frontend

A064617 a(n) = (10^n - 1)*(80/81) + n/9.

9, 98, 987, 9876, 98765, 987654, 9876543, 98765432, 987654321, 9876543210, 98765432099, 987654320988, 9876543209877, 98765432098766, 987654320987655, 9876543209876544, 98765432098765433, 987654320987654322, 9876543209876543211, 98765432098765432100, 987654320987654320989

Offset: 1

Henry Bottomley, Sep 26 2001

a(n)/10^n converges to 80/81 = 0.987654320987654320...

			Curious multiplications:
1*8 + 1 = 9;
12*8 + 2 = 98;
123*8 + 3 = 987;
1234*8 + 4 = 9876;
12345*8 + 5 = 98765;
123456*8 + 6 = 987654;
1234567*8 + 7 = 9876543;
12345678*8 + 8 = 98765432;
123456789*8 + 9 = 987654321.
- _Vincenzo Librandi_, Aug 07 2010 and _Philippe Deléham_, Mar 09 2014

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.

Harry J. Smith, Table of n, a(n) for n = 1..150
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275, A014824, A064616.

A064617:=n->(10^n-1)*(80/81)+n/9; seq(A064617(n), 1..20); # Wesley Ivan Hurt, Mar 10 2014

Table[(10^n - 1)*(80/81) + n/9, {n, 20}] (* Wesley Ivan Hurt, Mar 10 2014 *)
LinearRecurrence[{12,-21,10},{9,98,987},30] (* Harvey P. Dale, Aug 20 2023 *)

{ a=0; for (n=1, 150, a=10*a + 10 - n; write("b064617.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

Vec(x*(10*x-9)/((x-1)^2*(10*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = 10*a(n-1) + 10 - n = (19 - n)*A002275(n) - A064616(n) = 10*A002275(n) - A014824(n).
From Colin Barker, Sep 15 2014: (Start)
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3).
G.f.: x*(10*x - 9)/((x - 1)^2*(10*x - 1)). (End)
E.g.f.: exp(x)*(80*exp(9*x) + 9*x - 80)/81. - Stefano Spezia, May 28 2023

More terms from Colin Barker, Sep 15 2014

A099669 Partial sums of repdigits of A002276.

Original entry on oeis.org

2, 24, 246, 2468, 24690, 246912, 2469134, 24691356, 246913578, 2469135800, 24691358022, 246913580244, 2469135802466, 24691358024688, 246913580246910, 2469135802469132, 24691358024691354, 246913580246913576, 2469135802469135798, 24691358024691358020, 246913580246913580242

Offset: 1

Labos Elemer, Nov 17 2004

			2 + 22 + 222 + 2222 = a(4) = 2468.

Matthew House, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932.

A099669:=n->(2/81)*(10^(n+1) - 9*n - 10): seq(A099669(n), n=1..30); # Wesley Ivan Hurt, Apr 18 2017

<Robert G. Wilson v, Nov 20 2004 *)
LinearRecurrence[{12,-21,10},{2,24,246},30] (* Harvey P. Dale, Jun 01 2025 *)

a(n) = (2/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3). - Matthew House, Jun 30 2016
G.f.: 2*x/((1 - x)^2*(1 - 10*x)). - Ilya Gutkovskiy, Jun 30 2016
From Elmo R. Oliveira, Apr 02 2025: (Start)
E.g.f.: 2*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 2*A014824(n). (End)

A099675 Partial sums of repdigits of A002282.

Original entry on oeis.org

8, 96, 984, 9872, 98760, 987648, 9876536, 98765424, 987654312, 9876543200, 98765432088, 987654320976, 9876543209864, 98765432098752, 987654320987640, 9876543209876528, 98765432098765416, 987654320987654304, 9876543209876543192, 98765432098765432080, 987654320987654320968

Offset: 1

Labos Elemer, Nov 17 2004

			8 + 88 + 888 + 8888 + 88888 = a(5) = 98760.

Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Mathematica
```
<Robert G. Wilson v, Nov 20 2004 *)
```

a(n) = (8/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3). - Wesley Ivan Hurt, Jan 20 2024
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 8*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 8*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 8*A014824(n). (End)

More terms from Elmo R. Oliveira, Apr 02 2025

A287353 a(0)=0; for n>0, a(n) = 10*a(n-1) + prime(n).

Original entry on oeis.org

0, 2, 23, 235, 2357, 23581, 235823, 2358247, 23582489, 235824913, 2358249159, 23582491621, 235824916247, 2358249162511, 23582491625153, 235824916251577, 2358249162515823, 23582491625158289

Offset: 0

Luke Zieroth, May 23 2017

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Cf. A014824, A019518, A092844, A110299.

Cf. A379426 (prime terms).

FoldList[10 #1 + Prime@ #2 &, 0, Range@ 17] (* Michael De Vlieger, May 24 2017 *)

a(n) = fromdigits(primes(n)); \\ Kevin Ryde, Jun 22 2022

from sympy import prime
l = [0]
for i in range(20):
    l += [10 * l[i] + prime(i + 1)]
print(l) # Indranil Ghosh, May 25 2017

a(n) = Sum_{i=1..n} 10^(n-i)*prime(i), n >= 1. - Ya-Ping Lu, Dec 24 2024

A030512 Concatenation of first n 2-digit positive integers including leading zeros.

Original entry on oeis.org

1, 102, 10203, 1020304, 102030405, 10203040506, 1020304050607, 102030405060708, 10203040506070809, 1020304050607080910, 102030405060708091011, 10203040506070809101112, 1020304050607080910111213, 102030405060708091011121314

Offset: 1

Ralf Stephan

From Peter Bala, Sep 14 2015: (Start)
Empirically, we observe that the square roots of these numbers and their reciprocals have some interesting properties, as follows (examples are given below).
The decimal expansion of sqrt(a(n)) begins with strings of repeated digits (that gradually shorten in length until they disappear) alternating with strings of apparently random digits.
The decimal expansion of 1/sqrt(a(n)) has long strings of 0's (gradually shortening in length until they disappear) interspersed with blocks of digits. If we read these blocks of digits as ordinary integers and factorize them, we find the numbers are related in a surprising manner. Cf. A014824.
(End)

			From _Peter Bala_, Sep 14 2015: (Start)
Decimal expansions with repeating strings of digits in parentheses for clarity:
sqrt(a(50)) = 1.(0101...0101)0075(5050...5050)4728503 (7878...7878)7065734690(6565...6565)63090366531526199 (4949...4949)40423435587935014204(5454...5454) 511096186531728108723958(33...33)197004273464583079020182291 (66...66)107291492892700779438018798828124(99...99) 7645962810367893557912773556179470486(11...11) 010064064746152... * 10^49.
1/sqrt(a(10))  = 9.9(0...0)53955(0...0)441082125(0..0)4... * 10^(-10). The long strings of zeros gradually shorten in length until they disappear and are interspersed with five blocks of digits [99, 53955, 441082125, 400649596875, 38211955301953125] = [3^2*11, 3^2*5*11*109, 3^3*5^3*11*109^2, 3^2*5^5*11*109^3, 3^2*5^8*7*11*109^4].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..99 [Truncated to 99 terms by _Georg Fischer_, Mar 10 2020]

Cf. A007908, A014824, A262581, A262582.

[-(199/9801)-(1/99)*n+(10000/9801)*100^n: n in [0..98]]; // Vincenzo Librandi, May 17 2013

Table[-(199/9801)-(1/99) n + (10000/9801) 100^n, {n, 0, 98}] (* Vincenzo Librandi, May 17 2013 *)

a(n) = -(199/9801) - (1/99)*(n-1) + (10000/9801)*100^(n-1);
vector(20, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n+1) = 100*a(n) + n + 1 for n<100.

Edited by Charles R Greathouse IV, Apr 28 2010

Limits corrected by Georg Fischer, Mar 10 2020

A099674 Partial sums of repdigits of A002281.

Original entry on oeis.org

0, 7, 84, 861, 8638, 86415, 864192, 8641969, 86419746, 864197523, 8641975300, 86419753077, 864197530854, 8641975308631, 86419753086408, 864197530864185, 8641975308641962, 86419753086419739, 864197530864197516, 8641975308641975293, 86419753086419753070, 864197530864197530847

Offset: 0

Labos Elemer, Nov 17 2004

			7 + 77 + 777 + 7777 + 77777 = a(5) = 86415.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

<Robert G. Wilson v, Nov 20 2004 *)
Accumulate[LinearRecurrence[{11,-10},{0,7},25]] (* Harvey P. Dale, Jul 22 2025 *)

a(n) = (7/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 7*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 7*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 7*A014824(n).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 02 2025

a(0)=0 prepended by Harvey P. Dale, Jul 22 2025

A262183 a(0) = 0, a(n) = 10a(n-1) + n(n+1)*(n+2)/6.

Original entry on oeis.org

0, 1, 14, 150, 1520, 15235, 152406, 1524144, 15241560, 152415765, 1524157870, 15241578986, 152415790224, 1524157902695, 15241579027510, 152415790275780, 1524157902758616, 15241579027587129, 152415790275872430, 1524157902758725630, 15241579027587257840

Offset: 0

Peter Bala, Sep 14 2015

This sequence may be viewed as a generalization of A014824 and appears to share similar properties. See also A052262. We make the following empirical observations.
The decimal expansion of a(4*n+1)^(1/4) begins with strings of repeated digits (giving the appearance of rationality) alternating with strings of apparently random digits. The strings of repeated digits gradually shorten in length until they disappear and this pattern breaks down. See example (1) below. Brown calls numbers with these properties schizophrenic or mock-rational numbers. By increasing n we can prevent the disappearance of the repeating strings as long as we like. The repeating digits appear to always be an initial subsequence of [1, 3, 6, 7, 6, 3, 1, 9, 9, 3, 9, 9, 3, ....]. Cf. A060011.
It appears that the numbers (a(8*n+5))^(1/8) are also examples of Brown's schizophrenic numbers.
The decimal expansion of 1/(a(4*n+1))^(k/4) for k = 1,2,3,... has long strings of 0's (gradually shortening in length until they disappear) interlaced with blocks of digits. If we read these blocks of digits (for a fixed n and k) as ordinary integers and factorize them, we find the numbers are related in a surprising manner. See examples (2) through (5) below. Similar remarks apply to the decimal expansion of the numbers  1/(10*a(8*n))^(1/4), 1/(10*a(8*n+2))^(1/2), 1/(a(8*n+3))^(1/2), 1/(10*a(8*n+4))^(1/8), 1/(a(8*n+5))^(1/8), 1/(10*a(8*n+6))^(1/2), 1/(a(8*n+7))^(1/2) and their powers.
A theorem of Kuzmin in the measure theory of continued fractions says that for a random real number alpha, the probability that some given partial quotient of alpha is equal to a positive integer k is given by 1/log(2)*( log(1 + 1/k) - log(1 + 1/(k+1)) ). Thus large partial quotients are the exception in continued fraction expansions. Empirically, we observe the presence of unexpectedly large partial quotients early in the continued fraction expansions of the numbers (a(8*n+1))^(1/4), (a(8*n+3))^(1/2), (a(8*n+5))^(1/8), (a(8*n+7))^(1/2),
(10*a(8*n))^(1/4), (10*a(8*n+2))^(1/2), (10*a(8*n+4))^(1/8), (10*a(8*n+6))^(1/2) and their powers.

			(1) The decimal expansion of a(61)^(1/4) (with the blocks of 'random' digits enclosed in parentheses to aid readability) begins
1.111...111(026286308)333...333(2361974965884332291)666...666(4936365745813146737399105902)777...777(414516002742700195101894168058610026041)666...666(5834699239217156417791785081497321498627522786458)333..333(1... * 10^15.
The repeating digits are 1, 3, 6, 7, 6 and 3, an initial subsequence of A060011.
(2) The decimal expansion of 1/a(61)^(1/4) (with now the strings of 0's enclosed in parentheses) begins
9.(000..000)6870809025(000...000)131133379605615140625(000...000)300330802691003816294298046875(000...000)74515840736091563874877683318366943359375(000...000)193416219724333545001418899430083738351541748046875(000...000)5... * 10^(-16)
The long strings of 0's gradually shorten in length until they disappear and are interlaced with 5 strings of digits [6870809025, 131133379605615140625, 300330802691003816294298046875, 74515840736091563874877683318366943359375, 193416219724333545001418899430083738351541748046875]. Reading these strings as ordinary integers and factorizing we obtain [ (3^2)*(5^2)*30536929, (3^2)*(5^6)*(30536929)^2, (3^3)*(5^8)*(30536929)^3, (3^3)*(5^12)*13*(30536929)^4, (3^3)*(5^13)*13*17*(30536929)^3 ] showing how the numbers are related.
(3) The decimal expansion of 1/a(61)^(2/4) begins
8.1(000...000)12367456245(000...000)28324809994812870375(000...000)7207939264584091591063153125(000...000)192594788364052042015068473807471484375(000...000)52931280402387750233872466233174047490955859375(000...000)1... * 10^(-31).
The long strings of 0's gradually shorten in length and are interlaced with 5 strings of digits
[12367456245, 28324809994812870375, 7207939264584091591063153125, 192594788364052042015068473807471484375, 52931280402387750233872466233174047490955859375].
Reading these strings as ordinary integers and factorizing we obtain [ (3^4)*5*30536929, (3^5)*(5^3)*(30536929)^2, (3^4)*(5^5)*(30536929)^3, (3^4)*(5^8)*7*(30536929)^4, (3^6)*(5^8)*7*(30536929)^3 ].
(4) The decimal expansion of 1/a(61)^(3/4) begins
7.29(000...000)1669606593075(000...000)44611575741830270840625(000...000)124877547758919386815169127890625(000...000)35750407590077160299047085450511894287109375(000...000)1... * 10^(-46).
The long strings of 0's gradually shorten in length and are interlaced with 4 strings of digits [1669606593075, 44611575741830270840625, 124877547758919386815169127890625, 35750407590077160299047085450511894287109375]. Reading these strings as ordinary integers and factorizing we obtain [ (3^7)*(5^2)*30536929, (3^7)*(5^5)*7*(30536929)^2, (3^6)*(5^7)*7*11*(30536929)^3, (3^7)*(5^12)*7*11*(30536929)^4 ].
(5) The decimal expansion of 1/a(61) begins
6.561(000...000)200352791169(000...000)6118158958879580001(000...000)186829785738019654040356929(000...000)5705207902167118776034942675531041(000...000)174219528638716252198345946001761436313089(000...000)5320129376453944844526984070493622855630820563681(000...000)1... * 10^(-61).
The long strings of 0's gradually shorten in length and are interlaced with 6 strings of digits [200352791169, 6118158958879580001,  186829785738019654040356929, 5705207902167118776034942675531041, 174219528638716252198345946001761436313089, 5320129376453944844526984070493622855630820563681]. Reading these strings as ordinary integers and factorizing we obtain [ (3^8)*30536929, (3^8)*(30536929)^2, (3^8)*(30536929)^3, (3^8)*(30536929)^4, (3^8)*(30536929)^5, (3^8)*(30536929)^6 ].

Colin Barker, Table of n, a(n) for n = 0..1000
K. S. Brown, Mock-rational numbers.
Wikipedia, Schizophrenic Number
Index entries for linear recurrences with constant coefficients, signature (14,-46,64,-41,10).

Cf. A060011, A014824, A030512, A052262.

[0] cat [n eq 1 select 1  else 10*Self(n-1) + n*(n+1)*(n+2)/6: n in [1..30]]; // Vincenzo Librandi, Sep 20 2015

#A262183
seq((1/13122)*(2*10^(n+3)-243*n^3-1539*n^2-3096*n-2000), n = 0..22);

Table[(1/9^4) 10^(n + 3) - (243 n^3 + 1539 n^2 + 3096 n + 2000)/13122, {n, 0, 30}] (* Vincenzo Librandi, Sep 20 2015 *)
nxt[{n_,a_}]:={n+1,10a+((n+1)(n+2)(n+3))/6}; NestList[nxt,{0,0},20][[All,2]] (* or *) LinearRecurrence[ {14,-46,64,-41,10},{0,1,14,150,1520},30] (* Harvey P. Dale, Feb 29 2020 *)

concat(0, Vec(-x/((x-1)^4*(10*x-1)) + O(x^40))) \\ Colin Barker, Sep 20 2015

a(n) = (1/9^4)*10^(n+3) - (243*n^3 + 1539*n^2 + 3096*n + 2000)/13122.
O.g.f. x/((1 - 10*x)*(1 - x)^4).
a(n) = 14*a(n-1)-46*a(n-2)+64*a(n-3)-41*a(n-4)+10*a(n-5) for n>4. - Colin Barker, Sep 20 2015

A346535 Numbers obtained by adding the first k repdigits that consist of the same digit, for some number k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48, 60, 72, 84, 96, 108, 123, 246, 369, 492, 615, 738, 861, 984, 1107, 1234, 2468, 3702, 4936, 6170, 7404, 8638, 9872, 11106, 12345, 24690, 37035, 49380, 61725, 74070, 86415, 98760, 111105, 123456, 246912, 370368, 493824

Offset: 1

Jwalin Bhatt, Jul 22 2021

			a(1) = 1,
a(2) = 2,
a(3) = 3,
...
a(9) = 9;
a(10) = 1 + 11 = 12,
a(11) = 2 + 22 = 24,
a(12) = 3 + 33 = 36,
...
a(18) = 9 + 99 = 108;
a(19) = 1 + 11 + 111 = 123,
a(20) = 2 + 22 + 222 = 246,
a(21) = 3 + 33 + 333 = 369,
...
a(27) = 9 + 99 + 999 = 1107; ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,-21,0,0,0,0,0,0,0,0,10).

Cf. A010785, A014824.

Table[m*(10^(1+k)-10-9*k)/81,{k,6},{m,9}]//Flatten (* Stefano Spezia, Aug 17 2021 *)

def sumRepUnits(n): # A014824
  return ((10**n-1)*10 - 9*n)//81
def a(n): # A346535
  d = 1 + (n-1)%9
  m = 1 + (n-1)//9
  return d*sumRepUnits(m)
for n in range(1,1000):
  print(n, a(n))

a(n) = d*A014824(m) where d = (n-1) mod 9 + 1 and m = ceiling(n/9). - Jon E. Schoenfield, Jul 22 2021

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8)/((1 - x^9)^2*(1 - 10*x^9)). - Stefano Spezia, Jul 26 2021

A032343 a(n) = 10*a(n-1)+n^2, a(0)=0.

Original entry on oeis.org

0, 1, 14, 149, 1506, 15085, 150886, 1508909, 15089154, 150891621, 1508916310, 15089163221, 150891632354, 1508916323709, 15089163237286, 150891632373085, 1508916323731106, 15089163237311349, 150891632373113814

Offset: 0

Henry Bottomley, Jul 04 2000

a(n)/10^n converges to 110/729=0.15089163237311...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-33,31,-10).

Cf. A014824, A019521.

[0] cat [n le 1 select n else 10*Self(n-1)+n^2: n in [1..20]]; // Vincenzo Librandi, Mar 10 2013

CoefficientList[Series[(x^2+x)/((x-1)^3 (10x-1)),{x,0,30}],x]  (* Harvey P. Dale, Mar 20 2011 *)
RecurrenceTable[{a[0] == 0, a[n] == 10*a[n-1] + n^2}, a, {n, 100}] (* Vincenzo Librandi, Mar 10 2013 *)
LinearRecurrence[{13,-33,31,-10},{0,1,14,149},20] (* Harvey P. Dale, May 25 2024 *)

a(n) =(10^n-1)*(110/729)-n^2/9-n*(20/81).

G.f.: (x^2+x)/((x-1)^3*(10*x-1)). - Harvey P. Dale, Mar 20 2011

A099672 Partial sums of repdigits of A002279.

Original entry on oeis.org

5, 60, 615, 6170, 61725, 617280, 6172835, 61728390, 617283945, 6172839500, 61728395055, 617283950610, 6172839506165, 61728395061720, 617283950617275, 6172839506172830, 61728395061728385, 617283950617283940, 6172839506172839495, 61728395061728395050, 617283950617283950605

Offset: 1

Labos Elemer, Nov 17 2004

			5 + 55 + 555 + 5555 + 55555 = a(5) = 61725.

Colin Barker, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

<Robert G. Wilson v, Nov 20 2004 *)
Accumulate[Table[FromDigits[PadRight[{},n,5]],{n,0,20}]] (* Harvey P. Dale, Oct 05 2013 *)

Vec(5*x/((1 - x)^2*(1 - 10*x)) + O(x^40)) \\ Colin Barker, Nov 30 2017

a(n) = (5/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004.
From Colin Barker, Nov 30 2017: (Start)
G.f.: 5*x/((1 - x)^2*(1 - 10*x)).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 5*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 5*A014824(n). (End)

cp's OEIS Frontend

A064617 a(n) = (10^n - 1)*(80/81) + n/9.

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A099669 Partial sums of repdigits of A002276.

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Maple

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A099675 Partial sums of repdigits of A002282.

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Mathematica

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A287353 a(0)=0; for n>0, a(n) = 10*a(n-1) + prime(n).

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A030512 Concatenation of first n 2-digit positive integers including leading zeros.

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Magma

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A099674 Partial sums of repdigits of A002281.

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A262183 a(0) = 0, a(n) = 10*a(n-1) + n*(n+1)*(n+2)/6.

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A262183 a(0) = 0, a(n) = 10a(n-1) + n(n+1)*(n+2)/6.