A003618 -id:A003618 - cp's OEIS frontend

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

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

A133836 Largest prime with number of decimal digits equal to n-th prime.

Original entry on oeis.org

97, 997, 99991, 9999991, 99999999977, 9999999999971, 99999999999999997, 9999999999999999961, 99999999999999999999977, 99999999999999999999999999973, 9999999999999999999999999999973

Offset: 1

Parthasarathy Nambi, Jan 06 2008

			97 is the largest prime with 2 digits.
997 is the largest prime with 3 digits.
99991 is the largest prime with 5 digits.

Cf. A000040, A064490.

A003618 := proc(n) prevprime(10^n) ; end: A133836 := proc(n) A003618(ithprime(n)) ; end: for n from 1 to 13 do printf("%d,", A133836(n) ) ; od: # R. J. Mathar, Jan 30 2008

a(n)=A003618(A000040(n)). - R. J. Mathar, Jan 30 2008

More terms from R. J. Mathar, Jan 30 2008

A180927 Largest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

Original entry on oeis.org

8, 99, 994, 9994, 99997, 999994, 9999994, 99999994, 999999998, 9999999995, 99999999998, 999999999998, 9999999999998, 99999999999998, 999999999999995, 9999999999999998, 99999999999999998, 999999999999999987, 9999999999999999999

Offset: 1

Jonathan Vos Post, Jan 23 2011

This is to 3 and A014612, as 2 and A098450 (largest n-digit semiprime), and as 1 and A003618 (largest n-digit prime). Largest n-digit triprime. Largest n-digit 3-almost prime.

			a(1) = 8 because 8 = 2^3 is the largest (only) 1-digit number that is divisible by exactly 3 primes (counted with multiplicity).
a(2) = 99 because 99 = 3^2 * 11 is the largest 2-digit number (of 21) that is divisible by exactly 3 primes (counted with multiplicity).
a(3) = 994 because 994 = 2 * 7 * 71 is the largest 3-digit number that is divisible by exactly 3 primes (counted with multiplicity).

Cf. A003618, A014612, A098450, A180922.

lndn3[n_]:=Module[{k=10^n-1},While[PrimeOmega[k]!=3,k--];k]; Array[ lndn3,20] (* Harvey P. Dale, Jul 25 2019 *)

A180927(n)=forstep(n=10^n-1,10^(n-1),-1,bigomega(n)==3&return(n)) \\ M. F. Hasler, Jan 23 2011

A185201 10^n - second largest prime less than 10^n.

Original entry on oeis.org

5, 11, 9, 33, 11, 21, 27, 29, 71, 57, 53, 39, 137, 29, 53, 83, 23, 33, 57, 27, 113, 71, 53, 303, 321, 249, 107, 261, 53, 17, 81, 119, 47, 513, 237, 179, 123, 123, 173, 27, 203, 137, 119, 77, 119, 147, 83, 47, 183, 161, 333, 339, 611, 579

Offset: 1

Washington Bomfim, Jan 24 2012

nonn

			a(1) = 5 because precprime(10) = 7, and precprime(6) = 5.
From _M. F. Hasler_, Jul 19 2024: (Start)
Further examples: (where pp = prevprime = A151799)
    n |   pp(pp(10^n))  | a(n)
  ----+-----------------+------
    1 |               5 |   5
    2 |              89 |  11
    3 |             991 |   9
    4 |            9967 |  33
    5 |           99989 |  11
    6 |          999979 |  21
    7 |         9999973 |  27
    8 |        99999971 |  29
    9 |       999999929 |  71
   10 |      9999999943 |  57
   11 |     99999999947 |  53
   12 |    999999999961 |  39
   13 |   9999999999863 | 137
   14 |  99999999999971 |  29
   15 | 999999999999947 |  53
(End)

D. E. Knuth, The Art of Computer Programming, Second Edition, Vol. 2, Seminumerical Algorithms, Chapter 4.5.4 Factoring into Primes, Table 1, Page 390, Addison-Wesley, Reading, MA, 1981.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A033874.

Cf. A003618 (largest prime < 10^n), A151799 (prevprime function).

seq(10^n - prevprime(prevprime(10^n)),n=1..100); # Robert Israel, May 28 2017

Table[10^n - NextPrime[10^n, -2], {n,1,50}] (* G. C. Greubel, Jun 24 2017 *)

apply( {A185201(n)=10^n-precprime(precprime(10^n)-1)}, [1..66]) \\ M. F. Hasler, Jul 19 2024

a(n) = 10^n - precprime(precprime(10^n)-1)

A379140 Numbers k such that the greatest prime < 10^k and the least prime > 10^k share no decimal digits.

Original entry on oeis.org

1, 2, 8, 11, 15, 16, 17, 18, 21, 25, 26, 30, 40, 44, 46, 47, 50, 51, 53, 55, 60, 63, 64, 74, 77, 81, 86, 88, 89, 93, 95, 101, 123, 130, 131, 133, 134, 140, 152, 154, 158, 161, 164, 166, 176, 181, 189, 192, 198, 209, 214, 215, 233, 245, 264, 268, 274, 291, 293, 295, 297, 324, 326, 334, 352, 357

Offset: 1

Robert Israel, Dec 16 2024

Charles R Greathouse IV conjectures that A107801(n) = prime(n) for n sufficiently large (and similarly for other related sequences). If that is the case, this sequence must be finite.

			a(3) = 8 is a term because the greatest prime < 10^8 and the least prime > 10^8 are 99999989 and 100000007 respectively, and these have no digits in common.
5 is not a term because the greatest prime < 10^5 and the least prime > 10^5 are 99991 and 100003 respectively, and these have digit 1 in common.

Robert Israel, Table of n, a(n) for n = 1..250

Cf. A003617, A003618, A107801.

filter:= t -> convert(convert(prevprime(10^t),base,10),set) intersect convert(convert(nextprime(10^t),base,10),set) = {}:
select(filter, [$1..400]);

A068694 Largest n-digit prime with all odd digits.

Original entry on oeis.org

7, 97, 997, 9973, 99991, 999979, 9999991, 99999971, 999999937, 9999999557, 99999999977, 999999999959, 9999999999971, 99999999999973, 999999999999577, 9999999999999937, 99999999999999997, 999999999999999737

Offset: 1

Amarnath Murthy, Mar 03 2002

Cf. A068690, A068691, A068692, A068693.

Cf. A003618.

a(n)<=A003618(n). - R. J. Mathar, May 18 2007

More terms from Sascha Kurz, Mar 17 2002

A074489 n-th power of the largest n-digit prime.

Original entry on oeis.org

7, 9409, 991026973, 9892436613211441, 9995500809927103280440951, 999898004334901741252806480882137569, 9999937000170099744850229634875997137200865217031, 9999991200003387999254640102486990981144496037064410263414358881

Offset: 1

Zak Seidov, Sep 26 2002

			a(1)=7^1=7, a(2)=97^2=9409, a(3)=997^3=991026973.

Cf. A003618.

Mathematica
```
Table[Prime[PrimePi[10^n]]^n, {n, 6}]
```

a(n)=A003618(n)^n. - Emeric Deutsch, Dec 14 2004

More terms from Emeric Deutsch, Dec 14 2004

A097515 a(n) = (largest prime < 10^n) + (smallest prime > 10^n).

Original entry on oeis.org

18, 198, 2006, 19980, 199994, 1999986, 20000010, 199999996, 1999999944, 19999999986, 199999999980, 2000000000028, 20000000000008, 200000000000004, 2000000000000026, 19999999999999998, 200000000000000000, 1999999999999999992, 20000000000000000012, 200000000000000000028

Offset: 1

Cino Hilliard, Aug 27 2004

nonn

Cf. A003617, A003618.

Table[NextPrime[10^n,-1]+NextPrime[10^n],{n,20}] (* Harvey P. Dale, May 03 2018 *)

a(n) = A003618(n) + A003617(n+1). - Amiram Eldar, Jul 02 2024

More terms from Amiram Eldar, Jul 02 2024

A099657 a(n) is the least prime following A002277(n) repdigits.

Original entry on oeis.org

2, 5, 37, 337, 3343, 33343, 333337, 3333373, 33333347, 333333349, 3333333403, 33333333343, 333333333367, 3333333333347, 33333333333437, 333333333333389, 3333333333333343, 33333333333333391, 333333333333333391

Offset: 0

Labos Elemer, Nov 17 2004

			n=3: 33 is followed by 37.

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

Table[NextPrime[3*(10^n-1)/9], {n, 0, 35}]

A099661 a(n) is the least prime following A002281(n) repdigits.

Original entry on oeis.org

2, 11, 79, 787, 7789, 77783, 777781, 7777801, 77777803, 777777799, 7777777781, 77777777827, 777777777841, 7777777777859, 77777777777837, 777777777777787, 7777777777777867, 77777777777777797, 777777777777777817

Offset: 0

Labos Elemer, Nov 17 2004

			n=6: 77 is followed by 79.

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

Table[NextPrime[7*(10^n-1)/9], {n, 0, 35}]
NextPrime/@LinearRecurrence[{11,-10},{0,7},35] (* Harvey P. Dale, Dec 12 2021 *)

cp's OEIS Frontend

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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A133836 Largest prime with number of decimal digits equal to n-th prime.

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A180927 Largest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

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A185201 10^n - second largest prime less than 10^n.

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A379140 Numbers k such that the greatest prime < 10^k and the least prime > 10^k share no decimal digits.

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A068694 Largest n-digit prime with all odd digits.

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A074489 n-th power of the largest n-digit prime.

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A097515 a(n) = (largest prime < 10^n) + (smallest prime > 10^n).

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A099657 a(n) is the least prime following A002277(n) repdigits.