A033874 -id:A033874 - cp's OEIS frontend

A003618 Largest n-digit prime.

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

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

Original entry on oeis.org

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

A179975 Smallest k such that k*10^n is a sum of two successive primes.

Original entry on oeis.org

5, 3, 1, 6, 6, 6, 14, 6, 9, 19, 21, 21, 42, 93, 21, 6, 11, 2, 12, 111, 37, 39, 63, 38, 42, 24, 15, 15, 60, 6, 39, 82, 47, 58, 337, 49, 72, 25, 34, 21, 6, 107, 128, 96, 20, 2, 63, 231, 70, 7, 62, 144, 28, 151, 157, 33, 98, 55, 134, 162, 87, 201, 124, 303, 64, 106, 130, 13, 43

Offset: 0

Zak Seidov, Aug 04 2010

nonn

From Robert G. Wilson v, Aug 11 2010: (Start)
A179975 n's such that a(n)=1: 3, 335, ..., .
A179975 First occurrence of k: 3, 18, 2, ???, 1, 4, 50, 162, 9, 335, 17, 19, 68, 7, 27, ..., .
Records: 5, 6, 14, 19, 21, 42, 93, 111, 337, 449, 862, 1049, 1062, 1122, 1280, 2278, 3168, 4290, ..., . (End)

			a(0)=5 because 5=2+3
a(1)=3 because 30=13+17
a(2)=1 because 100=47+53
a(3)=6 because 6000=2999+3001.

Robert G. Wilson v, Table of n, a(n) for n = 0..400.
Dario Alejandro Alpern, Brilliant numbers

Cf. A064397, A071220, A074924, A074925.

Cf. A033873, A033874, A005235, A055211, A038804, A084475.

Join[{5,3},Reap[Do[Do[n=10^m k; If[n==PreviousPrime[n/2]+NextPrime[n/2],Sow[k];Break[]],{k,2000}],{m,2,50}]][[2,1]]]
f[n_] := Block[{k = 1, tn = 10^n}, While[h = k*tn/2; NextPrime[h, -1] + NextPrime@h != k*tn, k++ ]; k]; f[1] = 3; Array[f, 70, 0] (* Robert G. Wilson v, Aug 11 2010 *)

More terms from Robert G. Wilson v, Aug 11 2010

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

A350826 Number of prime sextuplets with n-digit initial term (A022008).

Original entry on oeis.org

1, 1, 0, 0, 3, 0, 13, 64, 235, 1296, 7013, 41782, 253420, 1607418, 10520883, 70785653, 488096844

Offset: 1

M. F. Hasler, Jan 17 2022

Prime sextuplets are of the form (p, p+4, p+6, p+10, p+12, p+16), where p is the initial member, listed in A022008.

For n = 1 and n = 2 (see Example), the last member of the sextuplet has one digit more than the initial member (so the count would be 0 for these two, if all terms of the sextuplet had to have the same length). As far as we know, for all n > 2, all members of the sextuplets have the same length. A sufficient condition for this is that A033874(n) > 16.

			For n = 1, p = 7 is the only 1-digit prime to be the initial term of a prime sextuplet, (7, 11, 13, 17, 19, 23), hence a(1) = 1.
For n = 2, p = 97 is the only 2-digit prime to be the initial term of a prime sextuplet, (97, 101, 103, 107, 109, 113), whence a(2) = 1.
For n = 3 and n = 4, there is no n-digit prime to be the initial term of a prime sextuplet, so a(n) = 0.
For n = 5, {16057, 19417, 43777} are the only 5-digit primes which are initial members of a prime sextuplet, therefore a(5) = 3.

Norman Luhn, PI_6(10^n)

Cf. A022008 (initial members of prime sextuplets), A033874 (10^n - precprime(10^n)).

Cf. A063501, A343636.

apply( {A350826(n,L=10^n)=n=L\10; for(c=0,oo, L<(n=next_A022008(n)) && return(c))}, [1..8])

a(n) = # { p in A022008 | 10^(n-1) < p < 10^n }.

a(10)-a(12) from David A. Corneth, Jan 17 2022

a(13)-a(17) from Hugo Pfoertner, Jan 21 2022

A104094 Largest prime <= 9^n.

Original entry on oeis.org

7, 79, 727, 6553, 59029, 531383, 4782961, 43046623, 387420479, 3486784393, 31381059607, 282429536453, 2541865828309, 22876792454939, 205891132094623, 1853020188851807, 16677181699666513, 150094635296999111

Offset: 1

Cino Hilliard, Mar 03 2005

Robert Israel, Table of n, a(n) for n = 1..1046

Cf. A013604.

Largest prime <= b^n: 2^n-A013603(n), 3^n-A013604(n), 4^n-A013606(n), 5^n-A013605(n), 6^n-A013607(n), 7^n-A013608(n), 8^n-A013603(3*n), 10^n-A033874(n).

f:= n -> prevprime(9^n):
map(f, [$1..30]); # Robert Israel, Aug 12 2019

NextPrime[#,-1]&/@(9^Range[20]) (* Harvey P. Dale, Apr 21 2024 *)

g(n,b) = for(x=0,n,print1(precprime(b^x)","))

a(n) = 9^n - A013604(2*n) = A001019(n) - A013604(2*n), n > 0. A.H.M. Smeets, Aug 12 2019

A157036 Shorthand for A157035, the largest prime with 2^n digits.

Original entry on oeis.org

3, 3, 27, 11, 63, 21, 51, 17, 813, 377, 7017, 27381, 7763, 1133, 119387, 67347, 121877

Offset: 0

Lekraj Beedassy, Feb 22 2009

The actual prime A157035(n) is obtained as 10^(2^n) - a(n).

Cf. A033874, A157034, A157035.

a:= n-> (t-> t-prevprime(t))(10^(2^n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Mar 02 2022

{ a(n) = 10^(2^n) - precprime(10^(2^n)) } \\ Max Alekseyev, Mar 28 2009

from sympy import prevprime
def a(n): return 10**(2**n) - prevprime(10**(2**n))
print([a(n) for n in range(10)]) # Michael S. Branicky, Mar 02 2022

a(n) = 10^(2^n) - A157035(n).

a(n) = A033874(2^n).

a(8)-a(13) from Ray Chandler and Max Alekseyev, Mar 22 2009
a(14) from Jinyuan Wang, Feb 22 2022
a(15) from Michael S. Branicky, Jun 19 2024
a(16) from Michael S. Branicky, Jun 27 2024

A038805 Difference between last prime < 10^n and 10^n is a record high.

Original entry on oeis.org

1, 4, 9, 21, 24, 34, 56, 66, 92, 100, 112, 117, 135, 180, 260, 349, 387, 393, 411, 574, 617, 787, 1209

Offset: 1

Jeff Burch

			For a(19)=411, the difference is 4433 and no other n <= 540 exceeds that difference.

Cf. A033874.

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; d = 0; k = 1; Do[ While[f = 10^k - PrevPrim[10^k]; d >= f, k++ ]; d = f; Print[k], {n, 1, 50}]

Edited and extended by Robert G. Wilson v, Jun 18 2002

a(23) from Robert G. Wilson v, Jan 10 2007

A097517 Consider the difference (A033873(k)) between the smallest prime > 10^k and 10^k; sequence lists the successive values of A033873(k) which are primes.

Original entry on oeis.org

7, 3, 3, 19, 7, 7, 19, 3, 37, 31, 37, 61, 3, 3, 7, 13, 67, 103, 331, 61, 193, 67, 43, 3, 109, 31, 193, 151, 31, 3, 19, 7, 79, 13, 103, 151, 373, 181, 31, 79, 97, 151, 127, 3, 3, 79, 3, 19, 457, 7, 139, 271, 79, 709, 79, 3, 283, 7, 283, 13, 13, 73, 67, 13, 151, 37, 193, 337

Offset: 1

Cino Hilliard, Aug 27 2004

nonn

This is the prime subsequence of A033873. Cf. A033874.

f[n_]:=Module[{x=10^n},NextPrime[x]-x];Select[f/@Range[0,200],PrimeQ] (* Harvey P. Dale, May 03 2021 *)

Definition corrected by N. J. A. Sloane, May 03 2021. Thanks to Harvey P. Dale for pointing out that something was wrong.

A118798 Numbers n such that the closest primes surrounding 10^n have the same last two digits.

Original entry on oeis.org

79, 178, 179, 186, 210, 284, 300, 349, 391, 456, 594, 595, 599, 624, 645, 654, 659, 704, 712, 713, 860, 871, 892, 904, 924, 990, 1015, 1089, 1097, 1110, 1118, 1151, 1165, 1374, 1396, 1459, 1709, 1721, 1826, 1831, 1911, 1943, 1956, 2005, 2061, 2082, 2089

Offset: 1

Cino Hilliard, May 23 2006

{251, 49}, 178 {239, 261}, 179 {221, 979}, 186 {479, 721}, 210 {171, 1129}, 284 {467, 133}, 300 {69, 331}, 349 {2603, 297}, 391 {123, 477}, 456 {633, 567}, 594 {11, 789}, 595 {503, 297}, 599 {2339, 2161}, 624 {413, 187}, 645 {3291, 109}, 654 {1811, 1089}, 659 {2363, 937}, 704 {3489, 211},
{171, 1029}, 713 {801, 2299}, 860 {1193, 2907}, 871 {827, 1473}, 892 {629, 271}, 904 {503, 597}, 924 {303, 4797}, 990 {3, 1197}, 1015 {71, 1029}, 1089 {4403, 5997}, 1097 {2271, 1429}, 1110 {2373, 2527}, 1118 {1767, 2233}, 1151 {2703, 97}, 1165 {33, 3867}, 1374 {689, 1411},
{1023, 3477}, 1459 {10211, 489}, 1709 {2859, 4241}, 1721 {10311, 189}, 1826 {1761, 1539}, 1831 {17751, 1449}, 1911 {4179, 2621}, 1943 {1279, 1721}, 1956 {541, 9459}, 2005 {141, 14259}, 2061 {6607, 3293}, 2082 {9537, 4563}, 2089 {597, 203}, 2091 {2517, 9783}, 2135 {7287, 3513}, ...,.

			79 is in the sequence since the two primes nearest primes 10^79 are 10^79 - 251 and 10^79 + 49.

V. Danilov, Smallest and largest n-digit primes.n

Cf. A115564.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ@k, k-- ]; k]; fQ[n_] := Mod[NextPrim[10^n], 100] == Mod[PrevPrim[10^n], 100]; Do[ If[ fQ@n, Print@n], {n, 2, 1250}] (* Robert G. Wilson v, May 27 2006 *)
Select[Range[2100],Mod[NextPrime[10^#],100]==Mod[NextPrime[10^#,-1],100]&] (* Harvey P. Dale, Mar 09 2019 *)

g(n) = for(j=1,n,x=precprime(10^j);y=nextprime(10^j);if(x%100==y%100,print1 (j",")))

A033873 + A033874 == 0 (mod 100). - Robert G. Wilson v, May 27 2006

More terms from Robert G. Wilson v, May 27 2006 - Jun 14 2006

cp's OEIS Frontend

A003618 Largest n-digit prime.

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

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A179975 Smallest k such that k*10^n is a sum of two successive primes.

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

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A350826 Number of prime sextuplets with n-digit initial term (A022008).

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A104094 Largest prime <= 9^n.

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A157036 Shorthand for A157035, the largest prime with 2^n digits.

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