A033873 -id:A033873 - cp's OEIS frontend

A127797 Nextprime(11^n)-11^n.

1, 2, 6, 30, 12, 2, 46, 20, 10, 2, 28, 62, 28, 42, 70, 30, 18, 20, 10, 18, 136, 102, 100, 30, 96, 6, 6, 68, 228, 30, 46, 48, 46, 32, 166, 36, 96, 42, 70, 278, 12, 108, 60, 42, 136, 68, 30, 18, 72, 36, 72, 30, 226, 252, 340, 126, 10, 42, 18, 182, 58, 18, 16, 120, 138, 36, 10

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127795, A127796, A127798, A127799.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[11^x] - 11^x; AppendTo[a, k], {x, 0, 100}]; a

A127798 Nextprime(12^n)-12^n.

Original entry on oeis.org

1, 1, 5, 5, 7, 7, 7, 25, 5, 31, 49, 31, 35, 25, 23, 11, 17, 29, 47, 103, 7, 5, 7, 23, 133, 19, 5, 13, 7, 215, 89, 5, 53, 89, 17, 35, 257, 29, 19, 193, 13, 121, 79, 71, 53, 61, 287, 61, 107, 125, 5, 203, 23, 119, 89, 5, 95, 61, 7, 29, 191, 211, 119, 31, 377, 37, 49, 89, 161, 5, 785

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127795, A127796, A127797, A127799.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[12^x] - 12^x; AppendTo[a, k], {x, 0, 100}]; a
f[n_]:=Module[{c=12^n},NextPrime[c]-c]; f/@Range[0,100]  (* Harvey P. Dale, Mar 19 2011 *)

A127799 Nextprime(13^n)-13^n.

Original entry on oeis.org

1, 4, 4, 6, 10, 6, 4, 6, 18, 46, 4, 34, 22, 16, 58, 4, 72, 28, 42, 34, 30, 166, 60, 16, 136, 46, 94, 66, 276, 30, 70, 136, 70, 18, 60, 142, 228, 10, 462, 12, 28, 166, 138, 12, 376, 16, 180, 102, 222, 228, 102, 126, 108, 46, 24, 172, 162, 6, 114, 6, 108, 6, 72, 84, 22, 70

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127795, A127796, A127797, A127798.

np13[n_]:=Module[{c=13^n},NextPrime[c]-c]; Array[np13,70,0] (* Harvey P. Dale, Mar 31 2012 *)

A157034 Shorthand for A157033, the smallest prime with 2^n digits.

Original entry on oeis.org

1, 1, 9, 19, 37, 33, 121, 283, 37, 241, 3259, 2823, 67017, 13989, 9523, 34281, 159007

Offset: 0

Lekraj Beedassy, Feb 22 2009

Cf. A033873, A157033, A157036.

a:= n-> (t-> nextprime(t)-t)(10^(2^n-1)):
seq(a(n), n=0..10);  # Alois P. Heinz, Mar 02 2022

f[n_] := NextPrime[ 10^(2^n-1)] - 10^(2^n-1); Table[ f@n, {n, 0, 12}] (* Robert G. Wilson v, Mar 17 2009 *)

from sympy import nextprime
def A157034(n): return 1 if n == 0 else nextprime(10**(2**n-1))-10**(2**n-1) # Chai Wah Wu, Apr 16 2021

a(n) = A157033(n) - 10^(2^n - 1).

a(8)-a(12) from Robert G. Wilson v, Mar 17 2009
a(13)-a(14) from Ray Chandler, Mar 22 2009
a(15) from Jinyuan Wang, Feb 24 2022
a(16) from Michael S. Branicky, Jun 18 2024

A096548 Difference between the smallest 10^n-digit prime and 10^(10^n-1).

Original entry on oeis.org

7, 289, 7, 33603, 309403, 593499

Offset: 1

Hugo Pfoertner, Jul 06 2004

Daniel Heuer found a(5) in 2004 by sieving up to 2^33 and then checking ~8000 candidates with pfgw-linux. Proving primality of 10^99999+309403 is beyond current (2004) technology.

a(6) was found by Kenneth Pedersen, Peter Kaiser, and Patrick De Geest. - Charles R Greathouse IV, Feb 11 2013

			a(1)=7 because the smallest ten-digit prime is 1000000007.
a(2)=289 because the smallest 100-digit prime is 10^99+289.

Prime Curios, 10000...33603 (10000-digits).
Chris K. Caldwell, The largest known primes.
Chris K. Caldwell, 10^999 + 7, related to a(3).
Chris K. Caldwell, 10^9999 + 33603, related to a(4).
factordb.com, 10^999 + 7, contains primality certificate related to a(3).
factordb.com, 10^9999 + 33603, contains primality certificate related to a(4).
Patrick De Geest, 10^999999 + y, history of a(6).
Henri Lifchitz, Renaud Lifchitz, Probable Primes Top 10000.
Henri Lifchitz, Renaud Lifchitz, 10^99999 + 309403, related to a(5).
Henri Lifchitz, Renaud Lifchitz, 10^999999 + 593499, related to a(6).
Hugo Pfoertner, Paul Underwood, Mike Oakes, Daniel Heuer, Smallest 100000-digit prime?, digest of 7 messages in primeform Yahoo group, Jul 8, 2004. [Cached copy]

Cf. A033873.

a(n) = nextprime(10^(10^n-1)) - 10^(10^n-1) = A007920(10^A002283(n)). - Jeppe Stig Nielsen, Jan 23 2021

A127795 Nextprime(8^n)-8^n.

Original entry on oeis.org

1, 3, 3, 9, 3, 3, 3, 17, 43, 29, 3, 17, 31, 23, 15, 59, 21, 21, 159, 9, 33, 29, 9, 29, 15, 33, 7, 17, 3, 39, 133, 105, 61, 255, 267, 39, 33, 51, 43, 29, 451, 165, 7, 17, 67, 33, 87, 5, 175, 51, 147, 95, 45, 299, 19, 141, 87, 129, 7, 75, 15, 215, 205, 35, 133, 35, 15, 351, 7, 203

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

"Nextprime(k)" is not well-defined: it can mean the smallest prime >= k or the smallest prime > k. Of course here it does not matter. - N. J. A. Sloane, Jan 31 2007

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127797, A127798, A127799.

np[n_]:=Module[{n8=8^n},NextPrime[n8]-n8]; Array[np,70,0] (* Harvey P. Dale, Jun 20 2011 *)

Erroneous Mathematica program deleted by Harvey P. Dale, Jun 20 2011

A175038 In the sequence of positive integers A000027, number of digits between successive primes.

Original entry on oeis.org

0, 1, 1, 4, 2, 6, 2, 6, 10, 2, 10, 6, 2, 6, 10, 10, 2, 10, 6, 2, 10, 6, 10, 14, 7, 3, 9, 3, 9, 39, 9, 15, 3, 27, 3, 15, 15, 9, 15, 15, 3, 27, 3, 9, 3, 33, 33, 9, 3, 9, 15, 3, 27, 15, 15, 15, 3, 15, 9, 3, 27, 39, 9, 3, 9, 39, 15, 27, 3, 9, 15, 21, 15, 15, 9, 15, 21, 9, 21, 27, 3, 27, 3, 15, 9, 15

Offset: 1

Zak Seidov, Nov 13 2009

From Jamie Morken, Feb 01 2019: (Start)
For A006880(m) < n < A006880(m+1), a(n) = A046933(n)*(m + 1).
For example m=1, n=24 then a(n)=7*2=14.
For example m=2, n=26 then a(n)=1*3=3.
For n = A006880(m+1), a(n) = A046933(n)*(m+1) + A033873(m + 1).
For example m=1, n=25 then a(n)=3*2+1=7.
(End)

			a(4) = 4 as prime(4) = 7 and prime(4+1) = 11 so the number of digits between these two primes is the number of digits of 8, 9 and 10. These numbers have 4 digits combined. Therefore a(4) = 4. - _David A. Corneth_, Jan 30 2019

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A000027, A113610, A046933, A006880, A033873.

Table[Length[Flatten[IntegerDigits/@Range[Prime[n]+1,Prime[n+1]-1]]],{n,200}]

a(n) = sum(k=prime(n)+1, prime(n+1)-1, #Str(k)); \\ Michel Marcus, Jan 30 2019

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

A260880 Smallest k such that 1000...0//k (concatenation of 10^n and k) is prime.

Original entry on oeis.org

1, 1, 9, 7, 3, 3, 37, 7, 7, 19, 3, 37, 31, 37, 61, 13, 3, 3, 39, 139, 57, 9, 49, 7, 67, 331, 319, 211, 57, 33, 49, 61, 103, 69, 67, 43, 321, 37, 3, 169, 63, 57, 31, 121, 9, 33, 217, 69, 9, 327, 171, 157, 31, 21, 279, 3, 193, 19, 67, 7, 121, 399, 57, 49, 49, 49

Offset: 0

Felix Fröhlich, Aug 02 2015

			The smallest prime whose decimal expansion starts with 10^6 = 1000000 is 100000037, so a(6) = 37.

Felix Fröhlich, Table of n, a(n) for n = 0..1000

Similar to but different from A033873.

a(n) = my(k=1); while(!ispseudoprime(eval(Str(10^n, k))), k++); k

A046445 Smallest composite with n prime factors that are distinct in length.

Original entry on oeis.org

1, 22, 2222, 2241998, 22435673986, 2243634705621958, 2243641436526074865874, 22436456994448042654162451606, 2243645856500003226552543739737161242, 2243645872205524222052566325604967420160128694

Offset: 0

Patrick De Geest, Jul 15 1998

Cf. A003617, A033873. Initial terms of A046442, A046443, A046444.

p = 2; Join[{1}, Table[p = p*Prime[PrimePi[10^n] + 1], {n, 9}]] (* Jayanta Basu, Jun 24 2013 *)

Corrected by Jayanta Basu, Jun 24 2013

cp's OEIS Frontend

A127797 Nextprime(11^n)-11^n.

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A127798 Nextprime(12^n)-12^n.

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A127799 Nextprime(13^n)-13^n.

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A157034 Shorthand for A157033, the smallest prime with 2^n digits.

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A096548 Difference between the smallest 10^n-digit prime and 10^(10^n-1).

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A127795 Nextprime(8^n)-8^n.

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A175038 In the sequence of positive integers A000027, number of digits between successive primes.

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Mathematica

PARI

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

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Mathematica

PARI

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Extensions

A260880 Smallest k such that 1000...0//k (concatenation of 10^n and k) is prime.

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