A033873 -id:A033873 - cp's OEIS frontend

A072098 Numbers k such that the difference between 10^k and the next prime > 10^k is a record high.

1, 3, 7, 12, 16, 21, 28, 62, 97, 118, 122, 135, 164, 218, 333, 346, 387, 443, 485, 521, 630, 819

Offset: 1

Robert G. Wilson v, Jun 18 2002

			For a(22)=819 the difference is 10443 and no other n <= 827 exceeds that difference.

Cf. A038805, A033873.

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

A078728 a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k10^n + m then k10^n+m is prime.

Original entry on oeis.org

3, 57, 297, 177, 237, 25111, 231339, 67419, 273817, 345111, 2001753, 912277, 5236153, 9228627, 10599391, 2835261, 60120003, 14054037, 27923629, 41783347, 24590943, 112161513, 230484021, 11446969, 205242589, 583389307, 873650007

Offset: 1

Farideh Firoozbakht, Dec 26 2003

nonn

a(n) is the smallest m such that m < 10^n and all six numbers 10^n + m, (Mod[m, 3]+2)*10^n + m, 4*10^n + m, (Mod[m, 3]+5)*10^n + m, 7*10^n + m & (Mod[m, 3]+8)*10^n + m are primes.

Carlos Rivera in Puzzle 245 of www.primepuzzles.net wrote "if the Faride's results ( a(n) for n=1,...,24 ) are plotted in Excel and a trend 'potential' function is asked, we obtain that a(n) is approximately equal to 0.5*n^6; this means that for n=999 a(n)=5*10^17, approximately." Since 10^n+a(n) is prime, for each n a(n)=0 (mod 3) or a(n)=1 (mod 3).

			a(6)=25111 because all the six numbers 1025111, 3025111, 4025111, 6025111, 7025111, 9025111 are primes and 25111 is the smallest number with this property.

Carlos Rivera, Puzzle 245. As 13

Cf. A003617, A033873.

Cf. A091199.

a[n_] := (For[m=1, !PrimeQ[10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+2)10^n+2m-1]||! PrimeQ[4*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+5)10^n+2m-1]||!PrimeQ [7*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+8)10^n+2m-1], m++ ];2m-1); Do[Print[a[n]], {n, 32}]

a[n_] := (For[m=1, !PrimeQ[10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+2)10^n+2m-1]||! PrimeQ[4*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+5)10^n+2m-1]||!PrimeQ [7*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+8)10^n+2m-1], m++ ];2m-1)

A097519 Prime differences between nextprime(2^n) and 2^n.

Original entry on oeis.org

3, 5, 3, 3, 7, 5, 3, 17, 3, 29, 3, 7, 17, 43, 29, 3, 11, 3, 11, 17, 53, 31, 7, 23, 29, 7, 59, 5, 5, 3, 131, 29, 13, 131, 3, 29, 11, 29, 37, 11, 7, 23, 13, 17, 3, 7, 29, 59, 61, 7, 277, 281, 43, 71, 29, 41, 277, 67, 7, 29, 17, 67, 37, 5, 5, 97, 7, 107, 19, 83, 7, 5, 107, 101

Offset: 1

Cino Hilliard, Aug 27 2004

nonn

Primes in A013597. - Bill McEachen, Oct 27 2021

Cf. A033873, A033874, A013597.

Select[Table[NextPrime[2^n]-2^n, {n,200}], PrimeQ] (* Harvey P. Dale, Dec 13 2011 *)
Select[NextPrime[#]-#&/@(2^Range[200]),PrimeQ] (* Harvey P. Dale, Jan 05 2024 *)

Definition corrected by Harvey P. Dale, Dec 13 2011

A120099 Numbers n such that the closest primes surrounding 10^n are the same distance modulo 100.

Original entry on oeis.org

17, 45, 87, 101, 112, 230, 270, 341, 468, 472, 473, 479, 517, 554, 555, 568, 650, 657, 663, 696, 718, 727, 810, 830, 836, 900, 917, 952, 984, 988, 1020, 1021, 1022, 1059, 1140, 1142, 1167, 1200, 1295, 1326, 1400, 1401, 1405, 1406, 1418, 1449, 1499, 1503, 1526

Offset: 1

Cino Hilliard and Robert G. Wilson v, Jun 08 2006

{3, 3}, 45 {9, 9}, 87 {373, 273}, 101 {3, 203}, 112 {207, 807}, 230 {753, 1053}, 270 {361, 861}, 341 {831, 1331}, 468 {301, 801}, 472 {1569, 2669}, 473 {99, 599}, 479 {109, 209}, 554 {937, 437}, 555 {151, 2151}, 568 {501, 801}, 650 {1999, 899}, 657 {1791, 291}, 663 {6333, 33},
{61, 1361}, 718 {5863, 1463}, 727 {273, 1073}, 810 {1591, 2891}, 830 {2853, 1253}, 836 {2809, 1209}, 900 {1873, 773}, 917 {693, 5393}, 952 {4827, 27}, 984 {1867, 2867}, 988 {753, 1053}, 1020 {793, 1193}, 1021 {1609, 6209}, 1022 {853, 1053} 1059 {5793, 1293}, 1140 {357, 4857},
{4329, 5829}, 1167 {1131, 3231}, 1200 {5227, 4127}, 1295 {5169, 2369}, 1326 {907, 4007}, 1400 {13317, 2517}, 1401 {10549, 2249}, 1405 {4329, 629}, 1406 {7477, 10277}, 1418 {841, 8741}, 1449 {2989, 3089}, 1499 {2001, 1901}, 1503 {439, 339}, 1526 {4603, 603}, 1534 {2409, 3209}, ...,.

Cf. A115564, A118798.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; PrevPrim[n_] := Block[{k = n - 1}, While[ ! PrimeQ@k, k-- ]; k]; Do[ If[ Mod[ NextPrim[10^n], 100] == Mod[10^n - PrevPrim[10^n], 100], Print[{n, a, b}]], {n, 1320}]
Select[Range[1530],Mod[NextPrime[10^#]-10^#,100]==Mod[10^# -NextPrime[ 10^#,-1],100]&] (* Harvey P. Dale, Sep 24 2021 *)

A033873 (mod 100) == A033874 (mod 100).

More terms from Robert G. Wilson v, Jun 09 2006

Corrected (term 517 added) by Harvey P. Dale, Sep 25 2021

A124050 Difference between (first Chen prime > 10^n) and 10^n.

Original entry on oeis.org

1, 1, 1, 9, 7, 19, 37, 19, 7, 7, 19, 19, 61, 687, 97, 91, 79, 13, 79, 97, 151, 217, 427, 253, 667, 13, 127, 427, 457, 577, 1069, 349, 1147, 1267, 2527, 2833, 709, 871, 259, 361, 1651, 391, 2689, 649, 31, 3007, 1657, 2257, 3757, 5977, 1441, 2779, 5749, 367, 31

Offset: 0

Zak Seidov, Nov 03 2006

nonn

A033873(n) <= a(n) <= A124001(n) and a(n) = A033873(n) for n = 0, 1, 2, 3, 4, 7, 8, 9, 10, 25, 44, 54.

			a(0) = 1 because 2 is prime, 2 + 2 = 4 semiprime and 2 - 10^0 = 1,
a(1) = 1 because 11 and 13 are twin primes and 11 - 10^1 = 1,
a(2) = 1 because 101 and 103 are twin primes and 101 - 10^2 = 1,
a(3) = 9 because 1009 is prime, 1011 = 3*337 semiprime and 1009 - 10^3 = 9,
a(4) = 7 because 10007 and 10009 are twin primes and 10007 - 10^4 = 7,
a(5) = 19 because 100019 is prime, 100021 = 29*3449 semiprime and 100019 - 10^5 = 19, etc.

Cf. A033873, A109611, A124001.

A214506 Last digit of the smallest prime number in base 10 with n digits.

Original entry on oeis.org

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

Offset: 1

Tjandra Satria Gunawan, Jul 19 2012

The last digit of the first prime number in base 10 with n digits for n = 1, 2, 3 is 211, because 2 is the least with 1 digit, 11 the least with 2 digits, 101 the least with 3 digits. The concatenation, 211, is prime. This does not happen again through 24 digits. - Jonathan Vos Post, Jul 04 2012

Table[Mod[NextPrime[10^n],10],{n,0,30}] (* Harvey P. Dale, Jan 25 2013 *)

a(1) = 2, and a(n) = A033873(n) mod 10 for n>1.

a(n) = A003617(n) mod 10. - Michel Marcus, Aug 06 2013

More terms from Harvey P. Dale, Jan 25 2013

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

cp's OEIS Frontend

A072098 Numbers k such that the difference between 10^k and the next prime > 10^k is a record high.

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A078728 a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k*10^n + m then k*10^n+m is prime.

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A097519 Prime differences between nextprime(2^n) and 2^n.

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A120099 Numbers n such that the closest primes surrounding 10^n are the same distance modulo 100.

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A124050 Difference between (first Chen prime > 10^n) and 10^n.

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A214506 Last digit of the smallest prime number in base 10 with n digits.

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

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A078728 a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k10^n + m then k10^n+m is prime.