A000042 -id:A000042 - cp's OEIS frontend

A083812 4n-1 is the digit reversal of n-1.

18, 198, 1998, 19998, 199998, 1999998, 19999998, 199999998, 1999999998, 19999999998, 199999999998, 1999999999998, 19999999999998, 199999999999998, 1999999999999998, 19999999999999998, 199999999999999998, 1999999999999999998, 19999999999999999998

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 08 2003

1. a(n) = 18 + 180 + 1800+ ...+ up to n terms. a(n) = sum of n terms of the geometric progression with the first term 18 and common ratio 10. 2. a(n) = 18*A000042(n).( the unary sequence).

			18 -1 = 17, 4*18 - 1 = 71.

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

Cf. A000042, A083811.

Accumulate[NestList[10#&,18,20]] (* or *) LinearRecurrence[{11,-10},{18,198},20] (* Harvey P. Dale, Apr 24 2015 *)

a(n) = 2*(10^n - 1).

a(1)=18, a(2)=198, a(n)=11*a(n-1)-10*a(n-2). - Harvey P. Dale, Apr 24 2015

Corrected and extended by Harvey P. Dale, Apr 24 2015

A087094 a(n) = smallest k such that (10^k-1)/9 == 0 mod prime(n)^2, or 0 if no such k exists.

Original entry on oeis.org

0, 9, 0, 42, 22, 78, 272, 342, 506, 812, 465, 111, 205, 903, 2162, 689, 3422, 3660, 2211, 2485, 584, 1027, 3403, 3916, 9312, 404, 3502, 5671, 11772, 12656, 5334, 17030, 1096, 6394, 22052, 11325, 12246, 13203, 27722, 7439, 31862, 32580, 18145, 37056, 19306

Offset: 1

Ray Chandler, Aug 10 2003

nonn

For a given a(n)>0, all of the values of k such that (10^k-1)/9=0 mod prime(n)^2 is given by the sequence a(n)*A000027, i.e. integral multiples of a(n). For example, for n=2, prime(2)=3, a(n)=9, the set of values of k for which (10^k-1)/9=0 mod 3^2 is 9*A000027=9,18,27,36,45,...
The union of the collection of sequences formed from the nonzero terms of a(n)*A000027, gives the values of k for which (10^k-1)/9 is not squarefree, see A046412. All of terms of the sequence a(n) are integer multiples of prime(n) for primes <1000 except for a(93)=486 where prime(93)=487. Conjecture: there are no 0 terms after a(3).
That conjecture is easily proved, for a(n) is just the multiplicative order of 10 modulo (prime(n))^2 for n>3. - Jeppe Stig Nielsen, Dec 28 2015

			a(2)=9 since 9 is least value of k for which (10^k-1)/9=0 mod 3^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000042, A046412, A084006, A084007.

0,9,0,seq(numtheory:-order(10,ithprime(i)^2), i=4..100); # Robert Israel, Dec 30 2015

a(n)=p=prime(n);10%p==0 && return(0);for(k=1,p^2,((10^k-1)/9) % p^2 == 0 && return(k));error() \\ Jeppe Stig Nielsen, Dec 28 2015

a(n)=p=prime(n);if(10%p==0, 0, 10%p==1, 9, znorder(Mod(10,p^2))) \\ Jeppe Stig Nielsen, Dec 28 2015

For n>3, a(n) = A084680(prime(n)^2) = A084680(A001248(n)), Jeppe Stig Nielsen, Dec 28 2015

A196104 Repdigit semiprimes (semiprimes composed of identical digits).

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 111, 1111, 11111, 1111111, 11111111111, 11111111111111111, 2222222222222222222, 3333333333333333333, 5555555555555555555, 7777777777777777777, 22222222222222222222222, 33333333333333333333333, 55555555555555555555555

Offset: 1

Michel Lagneau, Oct 27 2011

A semiprime can be repdigit (base 10) in only three ways. It can be a single-digit semiprime, a repunit semiprime (A102782), or a repunit prime times a prime digit {2, 3, 5, 7}. Occurs in proof that the sequence is infinite in which a(n) is the least semiprime > a(n-1) such that a(n) has no digit in common with a(n-1). - Jonathan Vos Post; corrected by Max Alekseyev, Sep 14 2022

			a(12) = 11111111111 = 21649 * 513239 is semiprime.

Max Alekseyev, Table of n, a(n) for n = 1..35

Subsequence of A046328.
Except for initial terms, subsequence of A116063.
Cf. A000042, A001358, A004023, A046413, A102782.

with(numtheory):for n from 1 to 23 do:for b from 1 to 9 do:x:=(((10^n)- 1)/9)*b:if bigomega(x)=2 then printf(`%d, `,x):else fi:od:od:

Select[FromDigits/@Flatten[Table[PadRight[{},i,n],{i,25},{n,9}],1], PrimeOmega[ #] ==2&] (* Harvey P. Dale, Mar 11 2019 *)

print1("4, 6, 9");for(n=1,20,t=10^n\9;if(bigomega(t)==2,print1(", "t)); if(isprime(t),forprime(p=2,7,print1(", "p*t)))) \\ Charles R Greathouse IV, Oct 27 2011

Union of {4, 6, 9}, A102782, 2*A004022, 3*A004022, 5*A004022, and 7*A004022. - Jonathan Vos Post and R. J. Mathar, Oct 27 2011

Edited by Max Alekseyev, Sep 14 2022

A258372 Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

Original entry on oeis.org

0, 3, 4, 8, 36, 8, 5, 72, 28, 6, 79, 212, 23, 6, 73, 24, 52, 62, 3, 28, 220, 53, 75, 58, 228, 9, 265, 89, 214, 86, 215, 4, 7, 39, 295, 40, 87, 216, 97, 6, 264, 53, 287, 223, 4, 239, 259, 25, 57, 364, 49, 38, 93, 86, 27, 30, 80, 24, 6, 356, 50, 645, 395, 206

Offset: 1

Felix Fröhlich, May 28 2015

n = 1 is the only case where a(n) = 0, since for any n > 1, A138148(n) is divisible by A002275(n).
No n exists such that a(n) = 2, since any number of the form A100706(n)+A011557(n) is of the form A000533(n)*A002275(n+1) (see comment by Robert Israel in A107123).
a(n) = 3 iff n is in A107123.
a(n) = 4 iff n is in A107124.
If k has an even number of digits and is a multiple of 11, then k is not a term. If k = (10^r+1)(10^m-1)/9 for some m > 0, r >= 0, then k is not a term. If A272232(k) = 0, then k is not a term. - Chai Wah Wu, Nov 08 2019

			a(1) = 0, because 101 is prime.
a(5) = 36, because the smallest x >= 0 such that 11111_x_11111 (where '_' denotes concatenation) is prime is 36. The decimal expansion of that prime is 111113611111.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A088281, A090287, A272232.

Table[k = 0; s = Table[1, {n}]; While[Or[!PrimeQ[FromDigits[s ~Join~ IntegerDigits[k] ~Join~ s]], Or[First@ IntegerDigits@ k == 1, Last@ IntegerDigits@ k == 1]], k++]; k, {n, 64}] (* Michael De Vlieger, May 28 2015 *)

a000042(n) = (10^n-1)/9
a(n) = my(k=0); while(k==10 || k%10==1 || k\(10^(#Str(k)-1))==1 || !ispseudoprime(eval(Str(a000042(n), k, a000042(n)))), k++); k

A261544 a(n) = Sum_{k=0..n} 1000^k.

Original entry on oeis.org

1, 1001, 1001001, 1001001001, 1001001001001, 1001001001001001, 1001001001001001001, 1001001001001001001001, 1001001001001001001001001, 1001001001001001001001001001, 1001001001001001001001001001001, 1001001001001001001001001001001001

Offset: 0

Ilya Gutkovskiy, Aug 24 2015

A sequence of palindromic numbers.

			From _Bruno Berselli_, Aug 25 2015: (Start)
a(n)   is the binary representation of    A023001
-------------------------------------------------
1  ...........................................  1
1001  ........................................  9
1001001 .....................................  73
1001001001  ................................  585
1001001001001  ............................  4681
1001001001001001  ........................  37449
1001001001001001001  ....................  299593
1001001001001001001001  ................  2396745
1001001001001001001001001  ............  19173961, etc.
(End)

Colin Barker, Table of n, a(n) for n = 0..333 (corrected by Michel Marcus, Jan 19 2019)
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019.
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1001,-1000).

Cf. A000533, A002113, A023001.
Subsequence of A033146.
Sums of 100^k: A094028; sums of 10^k: A000042.
Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

[(1000^(n+1)-1)/999: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015

Table[(1000^(n + 1) - 1)/999, {n, 0, 15}]
LinearRecurrence[{1001, -1000}, {1, 1001}, 20] (* Vincenzo Librandi, Aug 24 2015 *)

Vec(1 / ((x-1)*(1000*x-1)) + O(x^20)) \\ Colin Barker, Aug 24 2015

a(n) = (1000^(n + 1) - 1)/999.
a(n) = 1001*a(n-1) - 1000*a(n-2). - Colin Barker, Aug 24 2015
G.f.: 1 / ((x-1)*(1000*x-1)). - Colin Barker, Aug 24 2015
E.g.f.: (1/999)*(1000000*exp(1000*x) - exp(x)). - G. C. Greubel, Aug 29 2015

A362118 a(n) = (10^(n*(n+1)/2)-1)/9.

Original entry on oeis.org

1, 111, 111111, 1111111111, 111111111111111, 111111111111111111111, 1111111111111111111111111111, 111111111111111111111111111111111111, 111111111111111111111111111111111111111111111, 1111111111111111111111111111111111111111111111111111111, 111111111111111111111111111111111111111111111111111111111111111111

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Apr 19 2023

nonn

Concatenate 1, 11, 111, ..., 11...1 (n ones). There are n*(n+1)/2 1's in a(n).
This is a kind of unary analog of A058935, A360502, A117640, etc.
When regarded as decimal numbers, which (if any) is the smallest prime?
Answer: All terms > 1 are composite, since 111 is composite, all triangular numbers > 3 are composite and a prime repunit must have a prime number of decimal digits (see A004023). - Chai Wah Wu, Apr 19 2023. [This result was independently obtained by Michael S. Branicky, see A362429. - N. J. A. Sloane, Apr 20 2023]
a(45) has more than 1000 digits, and so cannot be included in the b-file. - Jason Bard, Apr 12 2025

			a(3) = 111111 because 3(3+1)/2 = 6, and 111111 has 6 ones.

Jason Bard, Table of n, a(n) for n = 1..44

Cf. A000042, A004023, A058935, A360502, A117640, A007908.

A362118[n_]:=(10^(n(n+1)/2)-1)/9;Array[A362118,10] (* Paolo Xausa, Nov 27 2023 *)

def A362118(n): return 10**(n*(n+1)>>1)//9 # Chai Wah Wu, Apr 19 2023

a(n) = A000042(A000217(n)). - Jason Bard, Apr 12 2025

A383241 a(n) = p(n)p(n+1)(p(n+1) - p(n)) - 1, where p(n) = prime(n).

Original entry on oeis.org

5, 29, 69, 307, 285, 883, 645, 1747, 4001, 1797, 6881, 6067, 3525, 8083, 14945, 18761, 7197, 24521, 19027, 10365, 34601, 26227, 44321, 69063, 39187, 20805, 44083, 23325, 49267, 200913, 66547, 107681, 38085, 207109, 44997, 142241, 153545, 108883, 173345

Offset: 1

Clark Kimberling, May 07 2025

nonn

			a(n) = A383242(n) - 2.

Cf. A000042, A383242, A383243, A383244.

z = 60; p[n_] := Prime[n];
f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
Table[f[n] - 1, {n, 1, z}]  (* A383241 *)
Table[f[n] + 1, {n, 1, z}]  (* A383242 *)

A383242 a(n) = p(n)p(n+1)(p(n+1) - p(n)) + 1, where p(n) = prime(n).

Original entry on oeis.org

7, 31, 71, 309, 287, 885, 647, 1749, 4003, 1799, 6883, 6069, 3527, 8085, 14947, 18763, 7199, 24523, 19029, 10367, 34603, 26229, 44323, 69065, 39189, 20807, 44085, 23327, 49269, 200915, 66549, 107683, 38087, 207111, 44999, 142243, 153547, 108885, 173347

Offset: 1

Clark Kimberling, May 07 2025

nonn

Cf. A000042, A383241, A383243, A383244.

z = 60; p[n_] := Prime[n];
f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
Table[f[n] - 1, {n, 1, z}]  (* A383241 *)
Table[f[n] + 1, {n, 1, z}]  (* A383242 *)

a(n) = A383241(n) + 2.

A383243 Primes of the form p(k)p(k+1)(p(k+1) - p(k)) - 1 sorted by increasing k.

Original entry on oeis.org

5, 29, 307, 883, 1747, 4001, 6067, 26227, 108883, 152083, 424481, 311347, 396883, 848201, 580627, 1713709, 1814509, 864883, 5092973, 3046789, 3386989, 1664083, 2581961, 2196307, 2304307, 2377747, 6955309, 3526867, 4088467, 20916053, 4796083, 7339361

Offset: 1

Clark Kimberling, May 07 2025

nonn

Conjecture: there are infinitely many such primes.

Cf. A000042, A383242, A383244.

Primes in A383241.

q:= 2; R:= NULL: count:= 0:
while count < 100 do
  p:= q;
  q:= nextprime(q);
  v:= p*q*(q-p)-1;
  if isprime(v) then R:= R,v; count:= count+1 fi;
od:
R; # Robert Israel, May 11 2025

z = 200; p[n_] := Prime[n];
f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
t1 = Table[f[n] - 1, {n, 1, z}];    (* A383241 *)
t2 = Table[f[n] + 1, {n, 1, z}];    (* A383242 *)
Select[t1, PrimeQ[#] &]  (* A383243 *)
Select[t2, PrimeQ[#] &]  (* A383244 *)

A383244 Primes of the form p(k)p(k+1)(p(k+1) - p(k)) + 1 sorted by increasing k.

Original entry on oeis.org

7, 31, 71, 647, 4003, 6883, 3527, 14947, 34603, 20807, 23327, 173347, 73727, 503869, 103967, 145799, 450403, 194687, 669283, 848203, 1193443, 1775563, 649799, 1976803, 2088547, 2131243, 4687069, 2534947, 2581963, 5338237, 3250123, 3411043, 1555847, 5346763

Offset: 1

Clark Kimberling, May 07 2025

nonn

Conjecture: there are infinitely many such primes.

Cf. A000042, A383241, A383243.

Primes in A383242.

=q:= 2; R:= NULL: count:= 0:
while count < 100 do
  p:= q;
  q:= nextprime(q);
  v:= p*q*(q-p)+1;
  if isprime(v) then R:= R,v; count:= count+1 fi;
od:
R; # Robert Israel, May 11 2025

z = 200; p[n_] := Prime[n];
f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
t1 = Table[f[n] - 1, {n, 1, z}];    (* A383241 *)
t2 = Table[f[n] + 1, {n, 1, z}];    (* A383242 *)
Select[t1, PrimeQ[#] &]  (* A383243 *)
Select[t2, PrimeQ[#] &]  (* A383244 *)

select(isprime, vector(200, k, prime(k)*prime(k+1)*(prime(k+1) - prime(k)) + 1)) \\ Michel Marcus, May 12 2025

cp's OEIS Frontend

A083812 4n-1 is the digit reversal of n-1.

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A087094 a(n) = smallest k such that (10^k-1)/9 == 0 mod prime(n)^2, or 0 if no such k exists.

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Maple

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A196104 Repdigit semiprimes (semiprimes composed of identical digits).

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Maple

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A258372 Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

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A261544 a(n) = Sum_{k=0..n} 1000^k.

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Magma

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A362118 a(n) = (10^(n*(n+1)/2)-1)/9.

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A383241 a(n) = p(n)*p(n+1)*(p(n+1) - p(n)) - 1, where p(n) = prime(n).

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A383241 a(n) = p(n)p(n+1)(p(n+1) - p(n)) - 1, where p(n) = prime(n).

A383242 a(n) = p(n)p(n+1)(p(n+1) - p(n)) + 1, where p(n) = prime(n).

A383243 Primes of the form p(k)p(k+1)(p(k+1) - p(k)) - 1 sorted by increasing k.

A383244 Primes of the form p(k)p(k+1)(p(k+1) - p(k)) + 1 sorted by increasing k.