A097166 -id:A097166 - cp's OEIS frontend

A246057 a(n) = (5*10^n - 2)/3.

1, 16, 166, 1666, 16666, 166666, 1666666, 16666666, 166666666, 1666666666, 16666666666, 166666666666, 1666666666666, 16666666666666, 166666666666666, 1666666666666666, 16666666666666666, 166666666666666666, 1666666666666666666, 16666666666666666666, 166666666666666666666

Offset: 0

Vincenzo Librandi, Aug 13 2014

a(k-1) = (10^k - 4)/6, together with b(k) = 3*a(k-1) + 2 = A093143(k) and c(k) = 2*a(k-1) + 1 = A002277(k) are k-digit numbers for k >= 1 satisfying the so-called curious cubic identity a(k-1)^3 + b(k)^3 + c(k)^3 = a(k)*10^(2*k) + b(k)*10^k + c(k) (concatenated a(k)b(k)c(k)). This k-family and the proof of the identity has been given in the introduction of the van der Poorten reference. Thanks go to S. Heinemeyer for bringing these identities to my attention. - Wolfdieter Lang, Feb 07 2017

			Curious cubic identities (see a comment and reference above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A curious cubic identity and self-similar sums of squares, The Mathematical Intelligencer, Vol. 29(2), pp. 39-41, March 2007.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. sequences with terms of the form 1k..k where the digit k is repeated n times: A000042 (k=1), A090843 (k=2), A097166 (k=3), A099914 (k=4), A099915 (k=5), this sequence (k=6), A246058 (k=7), A246059 (k=8), A067272 (k=9).

Cf. A002277, A086948, A093143, A323639.

Magma
```
[(5*10^n-2)/3: n in [0..20]];
```
Mathematica
```
Table[(5 10^n - 2)/3, {n, 0, 20}]
```

vector(50, n, (5*10^(n-1)-2)/3) \\ Derek Orr, Aug 13 2014

G.f.: (1 + 5*x)/((1 - x)*(1 - 10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
E.g.f.: exp(x)*(5*exp(9*x) - 2)/3. - Stefano Spezia, May 02 2025
a(n) = A323639(n+1)/2 = A086948(n+1)/12. - Elmo R. Oliveira, May 07 2025

A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 49, 56, 63, 70, 77

Offset: 1

Malo David, Nov 10 2021

The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Nov 23 2021

			For n=256, a(256) = 2*(2+5)*(2+5+6) = 182.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Malo David, About the Product Sum Digit Sequence

Cf. A055642, A284001 (binary analog), A349190 (fixed points).
Cf. A007953 (sum of digits), A059995 (floor(n/10)).
Cf. A349278 (similar, with the last digits).
Cf. A349898, A349899.

f:=func; [f(n):n in [1..100]]; // Marius A. Burtea, Nov 23 2021

Table[Product[Sum[Part[IntegerDigits[n],j],{j,i}],{i,Length[IntegerDigits[n]]}],{n,74}] (* Stefano Spezia, Nov 10 2021 *)

a(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ Michel Marcus, Nov 10 2021

first(n)=if(n<9,return([1..n])); my(v=vector(n)); for(i=1,9,v[i]=i); for(i=10,n, v[i]=sumdigits(i)*v[i\10]); v \\ Charles R Greathouse IV, Dec 04 2021

from math import prod
from itertools import accumulate
def a(n): return prod(accumulate(map(int, str(n))))
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 10 2021

For n>10: a(n) = a(A059995(n))*A007953(n) where A059995(n) = floor(n/10).
In particular, for n<100: a(n) = floor(n/10)*A007953(n)
From Bernard Schott, Nov 23 2021: (Start)
a(n) = 1 iff n = 10^k, k >= 0 (A011557).
a(n) = 2 iff n = 10^k + 1, k >= 0 (A000533 \ {1}).
a(n) = 3 iff n = 10^k + 2, k >= 0 (A133384).
a(n) = 5 iff n = 10^k + 4, k >= 0.
a(n) = 7 iff n = 10^k + 6, k >= 0. (End)
From Marius A. Burtea, Nov 23 2021: (Start)
a(A002275(n)) = n! = A000142(n), n >= 1.
a(A090843(n - 1)) = (2*n - 1)!! = A001147(n), n >= 1.
a(A097166(n)) = (3*n - 2)!!! = A007559(n).
a(A093136(n)) = 2^n = A000079(n).
a(A093138(n)) = 3^n = A000244(n). (End)

A070152 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives x of each pair.

Original entry on oeis.org

1, 2, 4, 13, 18, 33, 35, 7, 78, 133, 178, 228, 273, 388, 710, 1333, 1701, 1778, 2737, 3273, 3563, 3087, 3478, 12488, 13333, 14208, 17778, 31463, 36993, 5338, 7063, 9063, 12643, 15238, 17147, 22448, 23788, 27313, 29058, 34488, 36763, 38788, 43273, 50813, 53578

Offset: 1

Lekraj Beedassy, May 06 2002

From Bernard Schott, Jan 26 2022: (Start)
Some subsequences, from Diophante and Crux Mathematicorum:
{(2*10^m-5)/15, m >= 1} = 1, 13, 133, 1333, ... = A097166.
{2*(4*10^m+5)/45, m >= 1} = 2, 18, 178, 1778, ...
{13*(26*100^m-125)/12375, m >= 2} = 273, 27313, 2731313, ... (End)

			1+...+5 = 15; 2+...+7 = 27; 4+...+29 = 429; 13+...+53 = 1353; 18+...+63 = 1863.
133+...+533 = 133533.
273+...+2353 = 2732353.

Diophante, A1945 - Concaténations en tous genres (in French).
R. Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum 27(1), 2001, p. 39-44.

Cf. A070153, A071297, A186074.

Subsequence: A097166.

More terms from David W. Wilson, Jun 04 2002

Name edited by Michel Marcus, Jan 29 2022

A070153 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives y of each pair.

Original entry on oeis.org

5, 7, 29, 53, 63, 88, 91, 119, 403, 533, 623, 2148, 2353, 2813, 3835, 5333, 6076, 6223, 7889, 8728, 9163, 25039, 26603, 51513, 53333, 55168, 62223, 85338, 93633, 103463, 119063, 134938, 159518, 175238, 185979, 213073, 219413, 235313, 242818, 264888

Offset: 1

Lekraj Beedassy, May 06 2002

From Bernard Schott, Jan 26 2022: (Start)
Some subsequences, from Diophante and Crux Mathematicorum:
{(8*10^m-5)/15, m >= 1} = 5, 53, 533, 5333, ... (A350995).
{7*(4*10^m+5)/45, m >= 1} = 7, 63, 623, 6223, ...
{13*(224*100^m-125)/12375, m >= 2} = 2353, 235313, 23531313, ... (End)

			1+...+5 = 15; 2+...+7 = 27; 4+...+29 = 429; 13+...+53 = 1353; 18+...+63 = 1863.
133+...+533 = 133533.
178+...+623 = 178623.

Diophante, A1945 - Concaténations en tous genres (in French).
R. Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum 27(1), 2001, p. 39-44.

Cf. A070152, A071297, A097166, A186074.

Cf. A350995 (is a subsequence).

More terms from David W. Wilson, Jun 04 2002

Name edited by Michel Marcus, Jan 29 2022

A137233 Number of n-digit even numbers.

Original entry on oeis.org

5, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000

Offset: 1

Ctibor O. Zizka, Mar 08 2008

From Kival Ngaokrajang, Oct 18 2013: (Start)
a(n) is also the total number of double rows identified numbers in n digit.
For example:
  n = 1: 01 23 45 67 89 = 5 double rows;
  n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;
  n = 3: 100101 102103 104105...998999 = 450 double rows;
The number of double rows is also A030656. (End)
a(n) is also the number of n-digit integers with an even number of even digits (A356929); a(5) = 45000 is the answer to the question 2 of the Olympiade Mathématique Belge in 2004 (link). - Bernard Schott, Sep 06 2022
a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - Bernard Schott, Nov 07 2022

			a(2) = 45 because there are 45 2-digit even numbers.

Olympiade Mathématique Belge, OMB 2004, Finale Maxi, Question 2.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.

Cf. A030656, A052268, A054684, A356929.

Vec(5*x*(1-x)/(1-10*x) + O(x^22)) \\ Elmo R. Oliveira, Jul 23 2025

def A137233(n): return 9*10**(n-1)+1>>1 # Chai Wah Wu, Nov 11 2022

a(n) = 9*10^(n-1)/2 if n > 1. - R. J. Mathar, May 23 2008
From Elmo R. Oliveira, Jul 23 2025: (Start)
G.f.: 5*x*(1-x)/(1-10*x).
E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.
a(n) = 10*a(n-1) for n > 2.
a(n) = A052268(n)/2 for n >= 2. (End)

Corrected and extended by R. J. Mathar, May 23 2008

More terms from Elmo R. Oliveira, Jul 23 2025

A056698 Numbers k such that 10^k + 3*R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 15, 41, 83, 95, 341, 551, 669, 989, 1223, 6923, 103703

Offset: 1

Robert G. Wilson v, Aug 10 2000

Also numbers k such that (4*10^k-1)/3 is prime.

a(13) > 850000 (from Kamada data). - Robert Price, Oct 19 2014

Makoto Kamada, Prime numbers of the form 133...33.
Index entries for primes involving repunits

Cf. A002275, A093671, A097166, A259050.

Do[ If[ PrimeQ[ 10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 30470}]

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

a(12) from Kamada data by Robert Price, Oct 19 2014

A350995 a(n) = (16*10^n - 1)/3.

Original entry on oeis.org

5, 53, 533, 5333, 53333, 533333, 5333333, 53333333, 533333333, 5333333333, 53333333333, 533333333333, 5333333333333, 53333333333333, 533333333333333, 5333333333333333, 53333333333333333, 533333333333333333, 5333333333333333333, 53333333333333333333, 533333333333333333333, 5333333333333333333333

Offset: 0

Bernard Schott, Jan 28 2022

These terms 'y' form a subsequence of A070153 and the corresponding terms 'x' are in A097166 (see 3rd formula and examples).

			a(0) = (16-1)/3 = 5 and Sum_{j=1..5} = 15.
a(1) = (160-1)/3 = 53 and Sum_{j=13..53} = 1353.
a(2) = (1600-1)/3 = 533 and Sum_{j=133..533} = 133533.

Diophante, A1945 - Concaténations en tous genres (in French).
Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Subsequence of A070153.

Cf. A070152, A097166, A186074, A350994.

Maple
```
Data := seq((16*10^n-1)/3,  n = 0..21);
```

Table[(16*10^n - 1)/3, {n, 0, 21}] (* Amiram Eldar, Jan 29 2022 *)

a(n) = 10*a(n-1) + 3, n>0.
a(n) = 11*a(n-1) - 10*a(n-2), n>1.
A350994(n) = Sum_{j=A097166(n)..a(n)} = A097166(n).a(n) where "." means concatenation.
From Elmo R. Oliveira, May 02 2025: (Start)
G.f.: (5-2*x)/((1-x)*(1-10*x)).
E.g.f.: exp(x)*(16*exp(9*x) - 1)/3. (End)

A309907 a(n) is the square of the number consisting of one 1 and n 3's: (133...3)^2.

Original entry on oeis.org

1, 169, 17689, 1776889, 177768889, 17777688889, 1777776888889, 177777768888889, 17777777688888889, 1777777776888888889, 177777777768888888889, 17777777777688888888889, 1777777777776888888888889, 177777777777768888888888889, 17777777777777688888888888889

Offset: 0

Alois P. Heinz, Aug 21 2019

All terms are zeroless (elements of A052382).

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A052382, A097166, A102807.

a:= n-> parse(cat(1, 3$n))^2:
seq(a(n), n=0..18);

{a(n) = ((4*10^n-1)/3)^2} \\ Seiichi Manyama, Aug 23 2019

G.f.: -(40*x^2+58*x+1)/((x-1)*(100*x-1)*(10*x-1)).

a(n) = A097166(n)^2 = ((4*10^n-1)/3)^2. - Seiichi Manyama, Aug 23 2019

A350994 a(n) = (40100^n + 610^n - 1)/3.

Original entry on oeis.org

15, 1353, 133533, 13335333, 1333353333, 133333533333, 13333335333333, 1333333353333333, 133333333533333333, 13333333335333333333, 1333333333353333333333, 133333333333533333333333, 13333333333335333333333333, 1333333333333353333333333333, 133333333333333533333333333333

Offset: 0

Bernard Schott, Jan 28 2022

Subsequence of A186074.

Terms of this sequence satisfy the identity proposed in 2nd formula because a(n) = Sum_{j=(4*10^n-1)/3..(16*10^n-1)/3} j = ((4*10^n-1)/3).((16*10^n-1)/3) where "." means concatenation (see examples).

			a(0) = (40+6-1)/3 = Sum_{j=1..5} j = 15.
a(1) = (4000+60-1)/3 = Sum_{j=13..53} j = 1353.
a(2) = (400000+600-1)/3 = Sum_{j=133..533} j = 133533.

Diophante, A1945 - Concaténations en tous genres (in French).
Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A070152, A070153, A097166, A186074, A332167, A350995.

Data := seq((40*100^n + 6*10^n - 1)/3,  n = 0..17);

Table[(40*100^n + 6*10^n - 1)/3, {n, 0, 17}] (* Amiram Eldar, Jan 29 2022 *)
LinearRecurrence[{111,-1110,1000},{15,1353,133533},20] (* Harvey P. Dale, Jun 16 2025 *)

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3), n >= 3.
a(n) = Sum_{j=A097166(n)..A350995(n)} j = A097166(n).A350995(n) where "." means concatenation.
G.f.: (15 - 312*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Stefano Spezia, Jan 30 2022
a(n) = 2*A332167(n) + 1. - Hugo Pfoertner, Jan 30 2022
E.g.f.: exp(x)*(40*exp(99*x) + 6*exp(9*x) - 1)/3. - Elmo R. Oliveira, May 02 2025

A097169 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2)) * 3^k.

Original entry on oeis.org

1, 4, 13, 52, 133, 604, 1333, 6772, 13333, 74284, 133333, 801892, 1333333, 8550364, 13333333, 90286612, 133333333, 945912844, 1333333333, 9846548932, 13333333333, 101952273724, 133333333333, 1050903796852, 1333333333333

Offset: 0

Paul Barry, Jul 30 2004

a(n) = (4/3){1,10,10,100,100,1000...} -9{0,1,0,9,0,81...} -(1/3){1,1,1,1,1,1...} .
a(2n) = A097166(n).
a(2n+1)/4 = A097168(n).

Index entries for linear recurrences with constant coefficients, signature (1,19,-19,-90,90)

Cf. A097162, A075427.

LinearRecurrence[{1,19,-19,-90,90},{1,4,13,52,133},30] (* Harvey P. Dale, Dec 15 2017 *)

G.f.: (1+3x-10x^2-18x^3)/((1-x)*(1-9x^2)*(1-10x^2)).
a(n) = 2((1-sqrt(10))(-sqrt(10))^n+(1+sqrt(10))(sqrt(10))^n)/3+3((-3)^n-3^n)/2-1/3.
a(n) = a(n-1) +19a(n-2) -19a(n-3) -90a(n-4) +90a(n-5).

cp's OEIS Frontend

A246057 a(n) = (5*10^n - 2)/3.

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A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

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A070152 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives x of each pair.

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A070153 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives y of each pair.

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Author

Keywords

Comments

Examples

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Extensions

A137233 Number of n-digit even numbers.

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A056698 Numbers k such that 10^k + 3*R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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Mathematica

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A350995 a(n) = (16*10^n - 1)/3.

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A350994 a(n) = (40100^n + 610^n - 1)/3.