A003617 -id:A003617 - cp's OEIS frontend

A033873 Difference between first prime > 10^n (A003617) and 10^n.

1, 1, 1, 9, 7, 3, 3, 19, 7, 7, 19, 3, 39, 37, 31, 37, 61, 3, 3, 51, 39, 117, 9, 117, 7, 13, 67, 103, 331, 319, 57, 33, 49, 61, 193, 69, 67, 43, 133, 3, 121, 109, 63, 57, 31, 9, 121, 33, 193, 9, 151, 121, 327, 171, 31, 21, 3, 279, 159, 19, 7, 93, 447, 121, 57, 49, 49, 49, 99, 9

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net)

nonn

			The 23rd term is 117 since 10^23 + 117 = 100000000000000000000117 is prime and all of 100000000.., 100000....001, 1000...002 up through 1000...000116 are composite.

O'Hara, J. Rec. Math., 22 (1990), Table on page 278.

Matthias Baur, Table of n, a(n) for n = 0..14000 (terms 0..1000 from T. D. Noe, terms 1001..2750 from Robert G. Wilson v, terms 2751..8000 from Giovanni Resta)
Eric Weisstein's World of Mathematics, Next Prime
R. G. Wilson, v., Extract from letter to N. J. A. Sloane, May 20 1994, with annotated scanned copy of page 278 of O'Hara article.

Cf. A003617, A013632, A033874, A097517.

[NextPrime(10^n)-10^n: n in [0..65]]; // Vincenzo Librandi, Sep 13 2016

seq(nextprime(10^n)-10^n,n=0..70); # Emeric Deutsch, Apr 20 2006

f[n_]:=Module[{x=10^n},NextPrime[x]-x]; f/@Range[0,75]  (* Harvey P. Dale, Mar 30 2011 *)

a(n)=nextprime(10^n)-10^n \\ Charles R Greathouse IV, Jul 16 2011

a(n) = A003617(n) - 10^n = A013632(10^n). - Robert Israel, Aug 10 2015

More terms from Patrick De Geest

A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

Original entry on oeis.org

2, 1, 1, 1, 0, 1, 1, 0, 0, 9, 1, 0, 0, 0, 7, 1, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7, 1, 0, 0, 0

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A215641 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

Cf. A003617 (smallest n-digit prime), A215641 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340221 (same for semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

Flatten[Table[IntegerDigits[NextPrime[10^n]],{n,0,20}]] (* Harvey P. Dale, Mar 29 2024 *)

concat([digits(nextprime(10^k))|k<-[0..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*nextprime(10^(k-1))) \\ as constant

c = 0.21110110091000710000310000031000001910000000710000000071000000001...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*nextprime(10^(k-1))
a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

A272006 a(n) = A003617(n) - A062397(n-1).

Original entry on oeis.org

0, 0, 0, 8, 6, 2, 2, 18, 6, 6, 18, 2, 38, 36, 30, 36, 60, 2, 2, 50, 38, 116, 8, 116, 6, 12, 66, 102, 330, 318, 56, 32, 48, 60, 192, 68, 66, 42, 132, 2, 120, 108, 62, 56, 30, 8, 120, 32, 192, 8, 150, 120, 326, 170, 30, 20, 2, 278, 158, 18, 6, 92, 446, 120, 56, 48, 48, 48, 98, 8, 32, 272, 38, 78, 206

Offset: 1

Carauleanu Marc, Jul 13 2016

			For n=4, the smallest 4-digit prime is 1009, and 10^(4-1) + 1 = 1001, so a(4) = 1009 - 1001 = 8. - _Michael B. Porter_, Aug 01 2016

Cf. A003617, A033873, A062397.

Table[NextPrime[#] - (# + 1) &[10^(n - 1)], {n, 75}] (* Michael De Vlieger, Jul 13 2016 *)

a(n) = nextprime(10^(n-1)) - (10^(n-1) +  1); \\ Michel Marcus, Jul 28 2016

a(n) = A033873(n-1) - 1. - Michel Marcus, Jul 28 2016

A003618 Largest n-digit prime.

Original entry on oeis.org

7, 97, 997, 9973, 99991, 999983, 9999991, 99999989, 999999937, 9999999967, 99999999977, 999999999989, 9999999999971, 99999999999973, 999999999999989, 9999999999999937, 99999999999999997, 999999999999999989, 9999999999999999961, 99999999999999999989

Offset: 1

N. J. A. Sloane, Mira Bernstein

Since 10^n - 1 is always a multiple of 9, one could be tempted to think that 9 is the least frequently occurring least significant digit in terms of this sequence. - Alonso del Arte, Dec 03 2017

The occurrences of least significant digits in the first 8000 terms (see A033874) are 1: 2028, 3: 2032, 7: 2014, and 9: 1926. - Giovanni Resta, Mar 16 2020

			No power of 10 is prime.
9 = 3^2, 8 = 2^3 but 7 is prime, so a(1) = 7.
99 = 3^2 * 11 but 97 is prime, so a(2) = 97.
999 = 3^3 * 37 but 997 is prime, so a(3) = 997.
9999 = 3^2 * 11 * 101, 9997 = 13 * 769, 9995 = 5 * 1999, 9993 = 3 * 3331, 9991 = 97 * 103, ..., 9975 = 5^2 * 399, but 9973 is prime, so a(4) = 9973.

O'Hara, J. Rec. Math., 22 (1990), Table on page 278.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe. _Jon E. Schoenfield_ verified that terms 1..200 are indeed primes, Feb 02 2009)
Eric Weisstein's World of Mathematics, Previous Prime
R. G. Wilson, v., Extract from letter to N. J. A. Sloane, May 20 1994, with annotated scanned copy of page 278 of O'Hara article.

Cf. A003617, A033874, A114429 (largest n-digit twin prime).

[PreviousPrime(10^n): n in [1..20]]; // Vincenzo Librandi, Sep 13 2016

a:= n-> prevprime(10^n):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 11 2021

NextPrime[10^Range[20], -1] (* Harvey P. Dale, Feb 03 2011 *)

a(n)=precprime(10^n) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import prevprime
print([prevprime(10**n) for n in range(1, 21)]) # Michael S. Branicky, Feb 11 2021

More terms from Stefan Steinerberger, Apr 08 2006

A033874 Difference between the largest prime < 10^n (A003618) and 10^n.

Original entry on oeis.org

3, 3, 3, 27, 9, 17, 9, 11, 63, 33, 23, 11, 29, 27, 11, 63, 3, 11, 39, 11, 101, 27, 23, 257, 123, 141, 99, 209, 27, 11, 27, 21, 9, 411, 23, 159, 81, 59, 57, 17, 119, 83, 81, 53, 9, 33, 41, 33, 57, 57, 323, 231, 177, 291, 111, 593, 93, 149, 141, 161, 39, 83, 123, 51, 269

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net)

			a(4) = 27 because 10^4 - 9973 = 27. The 21st term is 101 since 10^21 - 101 = 999999999999999999899 is prime.

Knuth, Art of Computer Programming, volume 2, pages 13 and 390.
Journal of Recreational Mathematics, volume 14, number 4, page 285.
Journal of Recreational Mathematics, volume 20 ,number 3, page 209-210.
O'Hara, J. Rec. Math., 22 (1990), Table on page 278.

Giovanni Resta, Table of n, a(n) for n = 1..8000 (first 1000 terms from T. D. Noe)
V. Danilov, Table for large n
Eric Weisstein's World of Mathematics, Previous Prime
R. G. Wilson, v., Extract from letter to N. J. A. Sloane, May 20 1994, with annotated scanned copy of page 278 of O'Hara article.

Cf. A003618, A033873.

[10^n-PreviousPrime(10^n): n in [1..65]]; // Vincenzo Librandi, Sep 13 2016

seq(10^n-prevprime(10^n),n=1..65); # Emeric Deutsch, Apr 20 2006

PrevPrime[ n_Integer ] := Module[ {k}, k = n - 1; While[ ! PrimeQ[ k ], k-- ]; k ]; Table[ 10^n - PrevPrime[ 10^n ], {n, 1, 75} ] (* Robert G. Wilson v, Sep 09 2000 *)
Table[10^i - NextPrime[10^i, -1], {i, 0, 70}] (* Harvey P. Dale, Jan 13 2011 *)

a(n)=10^n-precprime(10^n) \\ Charles R Greathouse IV, Aug 03 2014

More terms from Patrick De Geest

A099656 a(n) is the least prime following A002276(n) repdigits.

Original entry on oeis.org

2, 3, 23, 223, 2237, 22229, 222247, 2222239, 22222223, 222222227, 2222222243, 22222222223, 222222222301, 2222222222243, 22222222222229, 222222222222227, 2222222222222281, 22222222222222301, 222222222222222281

Offset: 0

Labos Elemer, Nov 17 2004

			n=2: 22 is followed by 23.

Robert Israel, Table of n, a(n) for n = 0..999

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283.

seq(nextprime(2*(10^i-1)/9), i=0..20); # Robert Israel, Aug 25 2017

Table[NextPrime[2*(10^n-1)/9], {n, 0, 35}]
Table[ NextPrime[2*(10^n - 1)/9], {n, 0, 18}] (* Robert G. Wilson v, Nov 20 2004 *)
Table[NextPrime[FromDigits[PadRight[{},n,2]]],{n,0,20}] (* Harvey P. Dale, Dec 15 2021 *)

A215641 Smallest prime whose decimal expansion consists of the concatenation of a 1-digit prime, a 2-digit prime, a 3-digit prime, ..., and an n-digit prime.

Original entry on oeis.org

2, 211, 211151, 2111011129, 211101100910009, 211101100910007100049, 2111011009100071000031000453, 211101100910007100003100000310000721, 211101100910007100003100000310000019100000543, 2111011009100071000031000003100000191000000071000000531

Offset: 1

Jonathan Vos Post, Aug 18 2012

It is a plausible conjecture that a(n) always exists.

a(n) has A000217(n) = n*(n+1)/2 digits.

			a(4) = 2111011129, the smallest prime formed from a single-digit, a double-digit, a triple-digit, and a quadruple-digit prime, i.e., 2, 11, 101, 1129.

Cf. A000040, A000217, A003617.
Subsequence of A195302.
Cf. A338968 (similar, with largest prime).

Edited by N. J. A. Sloane, Aug 18 2012

A069588 Smallest prime in which the n-th significant digit is a 1.

Original entry on oeis.org

11, 11, 101, 1009, 10007, 100003, 1000003, 10000019, 100000007, 1000000007, 10000000019, 100000000003, 1000000000039, 10000000000037, 100000000000031, 1000000000000037, 10000000000000061, 100000000000000003, 1000000000000000003, 10000000000000000051

Offset: 1

Amarnath Murthy, Mar 25 2002

Essentially (i.e., except for the initial term), the same as A003617. The definition is misleading, since "n-th significant digit" seems to mean here "most significant digit" (except for a(1)), while the "significance" is decreasing when going from the first to the last digit. (E.g., 1234 rounded to 2 significant digits is 1200, so "1,2" should be the first and second (and not fourth and third) significant digits.) [M. F. Hasler, Jun 03 2009]

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Maple
```
11,seq(nextprime(10^j),j=1..30);
```

Join[{11}, NextPrime[10^Range[20]]] (* Paolo Xausa, Jun 23 2024 *)

from sympy import nextprime
def a(n): return 11 if n == 1 else nextprime(10**(n-1))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Nov 11 2021

More terms from Sascha Kurz, Mar 28 2002

a(19) and beyond from Michael S. Branicky, Nov 11 2021

A099668 a(n) is the largest prime before A002282(n) repdigits.

Original entry on oeis.org

7, 83, 887, 8887, 88883, 888887, 8888861, 88888883, 888888887, 8888888837, 88888888859, 888888888887, 8888888888857, 88888888888873, 888888888888883, 8888888888888753, 88888888888888801, 888888888888888859, 8888888888888888881, 88888888888888888879, 888888888888888888857

Offset: 1

Labos Elemer, Nov 17 2004

			n=2: 83 is before 88.

Cf. A002282, A003618, A003617, A007917, A096497, A096498, A068104, A002275-A002283, A099656-A099667.

Table[NextPrime[8*(10^n-1)/9, -1], {n, 1, 35}]
Table[NextPrime[FromDigits[PadRight[{},n,8]],-1],{n,20}] (* Harvey P. Dale, Jul 12 2014 *)

a(n) = A007917(A002282(n)). - Amiram Eldar, Jun 29 2025

A038804 Difference between largest n-digit prime and smallest (n+1)-digit prime.

Original entry on oeis.org

4, 4, 12, 34, 12, 20, 28, 18, 70, 52, 26, 50, 66, 58, 48, 124, 6, 14, 90, 50, 218, 36, 140, 264, 136, 208, 202, 540, 346, 68, 60, 70, 70, 604, 92, 226, 124, 192, 60, 138, 228, 146, 138, 84, 18, 154, 74, 226, 66, 208, 444, 558, 348, 322, 132, 596, 372, 308, 160, 168

Offset: 1

Jeff Burch

Records: 4, 12, 34, 70, 124, 218, 264, 540, 604, 670, 754, 1182, ..., . - Robert G. Wilson v, Jan 23 2020

			7 = greatest prime with 1 digit, 11 next smallest prime with 2 digits so a(1)=4.
97 = greatest prime with 2 digits, 101 next smallest prime with 3 digits so a(2)=4.

Giovanni Resta, Table of n, a(n) for n = 1..8000 (terms 1..1000 from Pierre CAMI, terms 1001..4000 from Robert G. Wilson v)
Pierre CAMI, PFGW script
Vasiliy Danilov, Smallest & largest n-digit primes.
Eric Weisstein's World of Mathematics, Next Prime.
Eric Weisstein's World of Mathematics, Previous Prime.

Cf. A033873, A033874, A003617, A003618, A058249.

(NextPrime[#]-NextPrime[#,-1])&/@(10^Range[100])  (* Harvey P. Dale, Mar 23 2011 *)

a(n) = A033873(n) + A033874(n). - Zak Seidov, Sep 13 2016

Corrected and edited by Patrick De Geest, Nov 06 2004

cp's OEIS Frontend

A033873 Difference between first prime > 10^n (A003617) and 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A272006 a(n) = A003617(n) - A062397(n-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A003618 Largest n-digit prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Extensions

A033874 Difference between the largest prime < 10^n (A003618) and 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A099656 a(n) is the least prime following A002276(n) repdigits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A215641 Smallest prime whose decimal expansion consists of the concatenation of a 1-digit prime, a 2-digit prime, a 3-digit prime, ..., and an n-digit prime.

Views

Author