A350995 -id:A350995 - cp's OEIS frontend

A097166 Expansion of g.f. (1+2x)/((1-x)(1-10*x)).

1, 13, 133, 1333, 13333, 133333, 1333333, 13333333, 133333333, 1333333333, 13333333333, 133333333333, 1333333333333, 13333333333333, 133333333333333, 1333333333333333, 13333333333333333, 133333333333333333, 1333333333333333333, 13333333333333333333, 133333333333333333333

Offset: 0

Paul Barry, Jul 30 2004

Partial sums of (1+2*x)/(1-10*x) = {1, 12, 120, 1200, ...}.
Second binomial transform of A082365.
These terms are the x's of A070152 and the corresponding y's are A350995 (see formula and examples). - Bernard Schott, Feb 15 2022

			a(0) = (4-1)/3 = 1 and Sum_{j=1..5} = 15.
a(1) = (40-1)/3 = 13 and Sum_{j=13..53} = 1353.
a(2) = (400-1)/3 = 133 and Sum_{j=133..533} = 133533.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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).

Cf. A056698 (index of primes), A082365, A097169, A309907 (squares of this).

Cf. A070152, A070153, A186074, A198970, A350994, A350995.

[(4*10^n-1)/3 : n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

a:= n-> parse(cat(1, 3$n)):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 23 2019

NestList[10#+3&,1,20] (* Harvey P. Dale, Jan 22 2014 *)

[(4*10**n-1)//3 for n in range(25)] # Gennady Eremin, Mar 04 2022

a(n) = (4*10^n - 1)/3.
a(n) = A097169(2*n).
a(n) = 10*a(n-1) + 3, n>0. a(n) = 11*a(n-1) - 10*a(n-2), n>1. - Vincenzo Librandi, Nov 01 2011
A350994(n) = Sum_{j=a(n)..A350995(n)} = a(n).A350995(n) where "." means concatenation. - Bernard Schott, Jan 28 2022
From Elmo R. Oliveira, Apr 29 2025: (Start)
E.g.f.: exp(x)*(4*exp(9*x) - 1)/3.
a(n) = A198970(n)/3. (End)

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

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

A056714 Numbers k such that 510^k + 3R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 3, 13, 25, 49, 143, 419, 1705, 13993, 35753, 40889

Offset: 1

Robert G. Wilson v, Aug 11 2000

nonn

Also numbers k such that (16*10^k - 1)/3 is prime.

5*10^a(n) + 3*(10^a(n) - 1)/9 is a solution for part (b) of questions of puzzle 244 from www.primepuzzles.net. If a(n) is greater than 5812 then a(n) is an example that is asked for in this question. - Farideh Firoozbakht, Dec 02 2003

Makoto Kamada, Prime numbers of the form 533...33.
Carlos Rivera, Puzzle 244. Null Conjunction, The Prime Puzzles and Problems Connection.
Index entries for primes involving repunits

Cf. A002275, A093674, A350995.

Do[ If[ PrimeQ[ 5*10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 5000}]

1705 from Farideh Firoozbakht, Dec 18 2003

13993, 35753 and 40889 from Serge Batalov, Jan 2009 confirmed as next terms by Ray Chandler, Feb 11 2012

cp's OEIS Frontend

A097166 Expansion of g.f. (1+2*x)/((1-x)*(1-10*x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A350994 a(n) = (40*100^n + 6*10^n - 1)/3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A056714 Numbers k such that 5*10^k + 3*R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A097166 Expansion of g.f. (1+2x)/((1-x)(1-10*x)).

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

A056714 Numbers k such that 510^k + 3R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.