A038804 -id:A038804 - cp's OEIS frontend

A331834 Records of A038804: (Smallest prime > 10^n) - (largest prime < 10^n).

4, 12, 34, 70, 124, 218, 264, 540, 604, 670, 754, 1182, 1310, 1418, 2606, 2702, 3516, 4248, 7358, 7680, 7692, 9060, 9962, 17568, 18052, 23784, 28218, 29814, 31024, 34788, 41438, 48984, 59568, 68986, 75944, 82700, 83232, 91840, 113108, 132652, 153094, 167608

Offset: 1

Robert G. Wilson v, Jan 28 2020

nonn

			Smallest 2-digit prime is 11, largest 1-digit prime is 7, so 11 - 7 = 4 is a term; since there is no 0-digit prime, 4 is a(1).

Cf. A038804.

Block[{s = ToExpression /@ Map[StringSplit, Drop[Import["https://oeis.org/A038804/b038804.txt", "Data"], 4]][[All, -1]]}, Union@ FoldList[Max, s]] (* Michael De Vlieger, Mar 14 2020, using b-file at A038804 *)
DeleteDuplicates[Table[NextPrime[10^n]-NextPrime[10^n,-1],{n,500}],GreaterEqual] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Aug 18 2025 *)

d=0;for(k=1,500,my(t=10^k,dd=nextprime(t)-precprime(t));if(dd>d,print1(dd,", ");d=dd)) \\ Hugo Pfoertner, Mar 01 2020

a(37)-a(42) from Giovanni Resta, Mar 15 2020

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

A058249 (Smallest prime >= 2^n) - (largest prime <= 2^n).

Original entry on oeis.org

0, 2, 4, 4, 6, 6, 4, 6, 12, 10, 14, 6, 18, 30, 22, 16, 30, 8, 22, 10, 26, 18, 24, 46, 74, 20, 68, 60, 14, 38, 12, 20, 26, 66, 84, 36, 34, 52, 30, 102, 48, 26, 86, 24, 114, 36, 120, 80, 150, 82, 150, 68, 116, 192, 58, 86, 22, 96, 186, 126, 16, 192, 54, 72, 180, 14, 22, 56

Offset: 1

Labos Elemer, Dec 05 2000

nonn

This sequence gives the gap between consecutive primes on either side of 2^n. The average gap between primes near 2^n should be about g=n*log(2). Cramer's conjecture would allow gaps to be as large as about g^2. - T. D. Noe, Jul 17 2007

			n = 1: a(1) = 2 - 2 = 0,
n = 9: a(9) = 521 - 509 = 12.

Robert G. Wilson v, Table of n, a(n) for n = 1..10087 (first 5000 terms from T. D. Noe).

Cf. A013603, A013597, A014210, A014234, A038804.

a := n -> if n > 1 then nextprime(2^n)-prevprime(2^n) else 0 fi; [seq( a(i), i=1..256)]; # Maple's next/prevprime functions use strict inequalities and therefore would not yield the correct difference for n=1. Alternatively, a(n) = nextprime(2^n-1)-prevprime(2^n+1);

Prepend[NextPrime[#]-NextPrime[#,-1]&/@(2^Range[2,70]),0] (* Harvey P. Dale, Jan 25 2011 *)
Join[{0}, Table[NextPrime[2^n] - NextPrime[2^n, -1], {n, 2, 70}]]

a(n)=nextprime(2^n)-precprime(2^n) \\ Charles R Greathouse IV, Sep 20 2016

a(n) = A014210(n) - A014234(n) = A013603(n) + A013597(n).

Edited by M. F. Hasler, Feb 14 2017

A338155 (Smallest prime >= 3^n) - (largest prime <= 3^n).

Original entry on oeis.org

0, 4, 6, 4, 10, 6, 24, 10, 6, 22, 36, 74, 30, 10, 18, 124, 44, 20, 70, 16, 60, 6, 52, 30, 34, 22, 42, 48, 144, 30, 20, 104, 122, 90, 50, 12, 52, 18, 140, 156, 72, 126, 126, 42, 68, 90, 98, 100, 66, 74, 50, 174, 30, 38, 126, 72, 30, 378, 102, 176, 108, 130

Offset: 1

A.H.M. Smeets, Oct 25 2020

nonn

Size of prime gap containing the number 3^n, for n > 1.

From Gauss's prime counting function approximation, the expected gap size should be approximately n*log(3), however, the observed values seem to be closer to n*log(8.72) ~ n*log(3^2) = n*A016632.

A.H.M. Smeets, Table of n, a(n) for n = 1..1000

Cf. A013598, A013604, A016632.

Cf. A058249 (for 2^n), A338419 (for 5^n), A338376 (for 6^n), A038804 (for 10^n).

a[1] = 0; a[n_] := First @ Differences @ NextPrime[3^n, {-1, 1}]; Array[a, 60] (* Amiram Eldar, Oct 30 2020 *)

a(n) = if (n==1, 0, nextprime(3^n) - precprime(3^n)); \\ Michel Marcus, Oct 25 2020

a(n) = A013598(n) + A013604(n) for n > 1.

A338376 (Smallest prime >= 6^n) - (largest prime <= 6^n).

Original entry on oeis.org

2, 6, 12, 6, 30, 14, 22, 18, 32, 12, 94, 54, 52, 18, 98, 66, 84, 18, 36, 18, 30, 138, 80, 96, 30, 142, 36, 80, 52, 26, 78, 64, 126, 138, 94, 136, 162, 276, 110, 162, 206, 94, 78, 324, 186, 128, 118, 56, 102, 390, 78, 90, 18, 62, 94, 108, 220, 100, 336, 618

Offset: 1

A.H.M. Smeets, Oct 26 2020

nonn

Size of prime gap containing the number 6^n, for n > 1.
From Gauss's prime counting function approximation, the expected gap size should be approximately n*log(6), however, the observed values seem to be closer to n*log(36) = n*A016659.
The arithmetic mean of a(n)/n for the terms 1..1000 is 3.605 ~ 2*log(6).

A.H.M. Smeets, Table of n, a(n) for n = 1..1000

Cf. A013600, A013607, A016659.

Cf. A058249 (2^n), A338155 (3^n), A338419 (5^n), A038804 (10^n).

a[n_] := First @ Differences @ NextPrime[6^n, {-1, 1}]; Array[a, 60] (* Amiram Eldar, Oct 30 2020 *)

a(n) = my(pw=6^n); nextprime(pw+1) - precprime(pw-1); \\ Michel Marcus, Oct 27 2020

a(n) = A013607(n) + A013600(n).

A338419 (Smallest prime >= 5^n) - (largest prime <= 5^n).

Original entry on oeis.org

0, 6, 14, 12, 16, 10, 16, 66, 42, 10, 26, 70, 58, 14, 46, 86, 18, 114, 72, 74, 78, 72, 74, 96, 78, 14, 50, 76, 78, 130, 110, 286, 164, 170, 424, 154, 70, 132, 336, 162, 160, 90, 400, 342, 144, 36, 208, 108, 284, 98, 138, 216, 20, 66, 132, 504, 320, 120, 354

Offset: 1

A.H.M. Smeets, Oct 25 2020

nonn

Size of prime gap containing the number 5^n, for n > 1.
From Gauss's prime counting function approximation, the expected gap size should be approximately n*log(5), however, the observed values seem to be closer to n*log(25) = n*A016648.
The arithmetic mean of a(n)/n for the terms 2..500 is 3.220 ~ 2*log(5) = A016648.

A.H.M. Smeets, Table of n, a(n) for n = 1..500
A.H.M. Smeets, Values of a(n)/n for n=2..500 ordered versus (n-1)/499

Cf. A013599, A013605, A016648.

Cf. A058249 (2^n), A338155 (3^n), A338376 (6^n), A038804 (10^n).

a[1] = 0; a[n_] := First @ Differences @ NextPrime[5^n, {-1, 1}]; Array[a, 60] (* Amiram Eldar, Oct 30 2020 *)

a(n) = if (n==1, 0, my(pw=5^n); nextprime(pw+1) - precprime(pw-1)); \\ Michel Marcus, Oct 27 2020

a(n) = A013599(n) + A013605(n) for n > 1.

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

A330468 Numbers k where the difference (least prime > 10^k) - (greatest prime < 10^k) sets a record.

Original entry on oeis.org

1, 3, 4, 9, 16, 21, 24, 28, 34, 66, 82, 92, 117, 122, 135, 218, 232, 314, 387, 443, 478, 617, 652, 787, 1031, 1157, 1440, 1625, 1872, 1920, 2038, 2424, 2692, 3235, 3331, 3798, 4944, 5202, 5241, 5938, 7572, 7847

Offset: 1

Hugo Pfoertner, Mar 01 2020

			a(1) = 1: 11 - 7 = 4;
a(2) = 3: 1009 - 997 = 12, whereas 101 - 97 = 4 <= a(1).

Cf. A003617, A003618, A038804, A331834.

d=0;for(k=1,500,my(t=10^k,dd=nextprime(t)-precprime(t));if(dd>d,print1(k,", ");d=dd))

More terms from Jinyuan Wang, Mar 01 2020

a(37)-a(42) from Giovanni Resta, Mar 15 2020

cp's OEIS Frontend

A331834 Records of A038804: (Smallest prime > 10^n) - (largest prime < 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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A058249 (Smallest prime >= 2^n) - (largest prime <= 2^n).

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A338155 (Smallest prime >= 3^n) - (largest prime <= 3^n).

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A338376 (Smallest prime >= 6^n) - (largest prime <= 6^n).

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A338419 (Smallest prime >= 5^n) - (largest prime <= 5^n).

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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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A330468 Numbers k where the difference (least prime > 10^k) - (greatest prime < 10^k) sets a record.

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