A003617 -id:A003617 - cp's OEIS frontend

A074490 n-th power of the smallest n-digit prime.

2, 121, 1030301, 1036488922561, 100350490343120066807, 1000180013500540012150145800729, 1000021000189000945002835005103005103002187, 100001520010108038410491224838661675728538309756103563041

Offset: 1

Zak Seidov, Sep 26 2002

			a(1)=2^1=2, a(2)=11^2=121, a(3)=101^3=1030301.

Vincenzo Librandi, Table of n, a(n) for n = 1..15

Table[Prime[PrimePi[10^(n-1)]+1]^n, {n, 7}]
#[[1]]^#[[2]]&/@Table[{NextPrime[10^n],n+1},{n,0,8}] (* Harvey P. Dale, Jan 24 2018 *)

a(n) = A003617(n)^n. - Michel Marcus, Oct 02 2013

a(8) from Vincenzo Librandi, Oct 04 2013

A099658 a(n) is the smallest prime greater than 4(10^n - 1)/9.

Original entry on oeis.org

2, 5, 47, 449, 4447, 44449, 444449, 4444469, 44444453, 444444457, 4444444447, 44444444497, 444444444461, 4444444444493, 44444444444459, 444444444444461, 4444444444444463, 44444444444444461, 444444444444444469, 4444444444444444537, 44444444444444444447

Offset: 0

Labos Elemer, Nov 17 2004

a(n) = smallest prime > A002278(n).

			n=4: 44 is followed by 47.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[4*(10^n-1)/9], {n, 0, 35}]

Checked by N. J. A. Sloane, Jan 27 2007

Mathematica program edited by Harvey P. Dale, Jul 16 2024

A099659 a(n) is the least prime following A002279(n) repdigits.

Original entry on oeis.org

7, 59, 557, 5557, 55579, 555557, 5555567, 55555559, 555555587, 5555555557, 55555555619, 555555555559, 5555555555593, 55555555555573, 555555555555557, 5555555555555573, 55555555555555639, 555555555555555559, 5555555555555555621, 55555555555555555567, 555555555555555555619

Offset: 1

Labos Elemer, Nov 17 2004

			n=2: 55 is followed by 59.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[5*(10^n-1)/9], {n, 1, 35}]

Offset corrected by Georg Fischer, Mar 12 2024

A099660 a(n) is the least prime following A002280[n] repdigits.

Original entry on oeis.org

2, 7, 67, 673, 6673, 66683, 666667, 6666679, 66666667, 666666667, 6666666757, 66666666667, 666666666671, 6666666666683, 66666666666737, 666666666666719, 6666666666666719, 66666666666666713, 666666666666666773

Offset: 0

Labos Elemer, Nov 17 2004

			n=6: 66 is followed by 67.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[6*(10^n-1)/9], {n, 0, 35}]
NextPrime/@Table[FromDigits[PadRight[{},n,6]],{n,0,20}] (* Harvey P. Dale, Feb 25 2015 *)

First Mathematica program corrected by Harvey P. Dale, Feb 25 2015

A099662 a(n) is the least prime following A002282(n) repdigits.

Original entry on oeis.org

2, 11, 89, 907, 8893, 88897, 888917, 8888927, 88888901, 888888901, 8888888891, 88888888901, 888888888919, 8888888888921, 88888888888889, 888888888888907, 8888888888888927, 88888888888888889, 888888888888888947

Offset: 0

Labos Elemer, Nov 17 2004

			n=8: 88 is followed by 89.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[8*(10^n-1)/9], {n, 0, 35}]

A099664 a(n) is the largest prime before A002278(n).

Original entry on oeis.org

3, 43, 443, 4441, 44417, 444443, 4444409, 44444399, 444444443, 4444444429, 44444444441, 444444444443, 4444444444439, 44444444444353, 444444444444421, 4444444444444423, 44444444444444411, 444444444444444419

Offset: 1

Labos Elemer, Nov 17 2004

			n=2: 43 is before 44.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Mathematica
```
<Harvey P. Dale, Feb 25 2013 *)
```

A101592 Smallest and largest of the n-digit primes.

Original entry on oeis.org

2, 7, 11, 97, 101, 997, 1009, 9973, 10007, 99991, 100003, 999983, 1000003, 9999991, 10000019, 99999989, 100000007, 999999937, 1000000007, 9999999967, 10000000019, 99999999977, 100000000003, 999999999989, 1000000000039, 9999999999971, 10000000000037

Offset: 1

Parthasarathy Nambi, Jan 26 2005

			2 is the smallest 1-digit prime and 7 is the largest 1-digit prime.
100003 is the smallest 6-digit prime and 999983 is the largest 6-digit prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Chris K. Caldwell List of small primes.

Cf. A003617, A003618

Table[{NextPrime[10^n],NextPrime[10^(n+1),-1]},{n,0,15}]//Flatten (* Harvey P. Dale, Jun 04 2019 *)

A109937 Number of consecutive composite numbers in successive consecutive number sets in A109936; (smallest (n+1)-digit prime) - (largest n-digit prime) - 1.

Original entry on oeis.org

3, 3, 11, 33, 11, 19, 27, 17, 69, 51, 25, 49, 65, 57, 47, 123, 5, 13, 89, 49, 217, 35, 139, 263, 135, 207, 201, 539, 345, 67, 59, 69, 69, 603, 91, 225, 123, 191, 59, 137, 227, 145, 137, 83, 17, 153, 73, 225, 65, 207, 443, 557, 347, 321, 131, 595, 371, 307, 159, 167

Offset: 1

Amarnath Murthy, Jul 19 2005

Cf. A109936.

A003617 := proc(n) nextprime(10^(n-1)) ; end: A003618 := proc(n) prevprime(10^n) ; end: A038804 := proc(n) A003617(n+1)-A003618(n) ; end: A109937 := proc(n) A038804(n)-1 ; end: seq(A109937(n),n=1..60) ; # R. J. Mathar, Feb 08 2008

a(n) = A038804(n)-1. - R. J. Mathar, Feb 08 2008

More terms from R. J. Mathar, Feb 08 2008

A141116 Smallest n-digit prime with no identical adjacent digits (or 0 if no such prime exists).

Original entry on oeis.org

2, 13, 101, 1013, 10103, 101021, 1010129, 10101023, 101010157, 1010101039, 10101010163, 101010101063, 1010101010131, 10101010101019, 101010101010131, 1010101010101037, 10101010101010141, 101010101010101083

Offset: 1

Rick L. Shepherd, Jun 05 2008

For n >= 1, a(n) >= A056830(n), the least n-digit positive integer with no identical adjacent digits (also the least positive integer whose digits occur in n runs). Conjecture: For all n, a(n) <> 0.
If the conjecture is true, then this sequence and the following two sequences are equivalent: i) Smallest prime with exactly n runs of digits and ii) Smallest prime with at least n runs of digits. For each n <= 625, a(n) is an n-digit prime (provided that each probable prime shown in the link is indeed a prime -- or at least one of very many (slightly) larger probable prime candidates is prime).
As each a(n) shown is very near A056830(n), I believe it is extremely unlikely that a randomly-given n would yield a 0 term (but I don't have a proof for arbitrary n).

			a(4) = 1013 because 1013 is the smallest 4-digit prime having no identical adjacent digits; the only smaller 4-digit prime, 1009, is disqualified by the "00", identical adjacent digits (of run length 2). Also each digit, 1, 0, 1, 3, occurs in a run of identical digits of length 1 for a total of 4 runs with 1013 being the smallest prime of any length with 4 runs of digits.

Rick L. Shepherd, Table of n, a(n) for n = 1..625

Cf. A003617, A007809, A056830.

A262083 Smallest possible prime factor of 10^k+n for any k.

Original entry on oeis.org

2, 7, 2, 7, 2, 3, 2, 17, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 11, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 13, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 17, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 17, 2, 7, 2, 3, 2, 7, 2, 7, 2

Offset: 0

Sergio Pimentel, Sep 10 2015

nonn

Is this sequence bounded?  What are the records for a(n)?
From Robert G. Wilson v, Sep 13 2015: (Start)
First occurrence of the i-th prime: 0, 5, 15, 1, 21, 49, 7, 357, 24871, 364021, ..., .
a(n) = 2 when n == 0 (mod 2),
a(n) = 3 when n == 5 (mod 6),
a(n) = 5 when n == 15 or 25 (mod 30),
a(n) = 7 when n == 1, 3, 9, 13, 19, 27, 31, 33, 37, 39, 43, 51, 57, 61, 67, 69, 73, 79, 81, 87, 93, 97, 99, 103, 109, 111, 117, 121, 123, 127, 129, 139, 141, 151, 153, 157, 159, 163, 169, 171, 177, 181, 183, 187, 193, 199, 201 or 207 (mod 210),
a(n) = 11 when n = 21, 133, 441, 483, 637, 903, 1057, 1099, 1407, 1519, 1561, 1827, 1869, 1981, 2023 or 2289 (mod 2310),
a(n) = 13 when n = 49, 147, 217, 231, 259, 399, 469, 511, 651, 679, 693, 763, 777, 861, 987, 1141, 1197, (413 terms missing), 29883 or 29953, ... (mod 30030),
a(n) = 17 when n = 7, 63, 91, 189, 273, 301, 343, 427, 553, 567, 609, 721, 819, 847, 889, 931, 973, 1029, (8044 terms missing), 510349 or 510447 (mod 510510),
a(n) = 19 when n = 357, 1071, 2737, 3451, 6069, 6307, 8211, 9163, 9639, 10353, 12019, 12733, 13447, 13923, 15351, 15589, 17017, 17493, 18207, ... (mod 9699690),
a(n) = 23 when n = 24871, 47481, 74613, 88179, 92701, 106267, 133399, 142443, 160531, 187663, 201229, 210273, 223839, 250971, 264537, 309757, ... (mod 223092870),
a(n) = 29 when n = 364021, 988057, ... (mod 6469693230), etc.
To the question if this sequence is 'bounded', I would answer no.
(End)
For complete lists of when a(n) < 19, see Wilson's Congruencies a-file. - Danny Rorabaugh, Oct 08 2015

			a(1) = 7 since 10^k+1 is not divisible by 2,3 or 5 for all k but is divisible by 7 when k = 3 (i.e., 1001 = 7*11*13).

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Robert G. Wilson v, Congruencies for A262083

Cf. A000533, A003617, A038371, A185121.

p = Prime@ Range@ 25; f[n_] := Block[{k = 1, lst = {}}, While[k < 25, AppendTo[lst, Position[ Mod[ PowerMod[10, k, p] + n, p] 0, 1, 1][[1, 1]]]; k++]; lst = Union@ lst; Prime@ lst[[1]]]; Array[f, 101, 0] (* Robert G. Wilson v, Sep 13 2015 *)

More terms from Robert G. Wilson v, Sep 13 2015

cp's OEIS Frontend

A074490 n-th power of the smallest n-digit prime.

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A099658 a(n) is the smallest prime greater than 4(10^n - 1)/9.

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A099659 a(n) is the least prime following A002279(n) repdigits.

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A099660 a(n) is the least prime following A002280[n] repdigits.

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A099662 a(n) is the least prime following A002282(n) repdigits.

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A099664 a(n) is the largest prime before A002278(n).

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Mathematica

A101592 Smallest and largest of the n-digit primes.

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Mathematica

A109937 Number of consecutive composite numbers in successive consecutive number sets in A109936; (smallest (n+1)-digit prime) - (largest n-digit prime) - 1.

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A141116 Smallest n-digit prime with no identical adjacent digits (or 0 if no such prime exists).

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