A058249 -id:A058249 - cp's OEIS frontend

A331620 Records of A058249: (Smallest prime >= 2^n) - (largest prime <= 2^n).

2, 4, 6, 12, 14, 18, 30, 46, 74, 84, 102, 114, 120, 150, 192, 242, 306, 370, 414, 570, 790, 800, 888, 924, 1024, 1152, 1310, 1790, 2262, 2754, 3472, 6270, 6458, 9334, 12628, 14200, 16406, 18050, 18358, 18492, 19852, 23180, 28730, 32592, 33524, 38696, 39774, 40944, 45444, 46408, 49624, 54704, 60740

Offset: 1

Robert G. Wilson v, Jan 22 2020

nonn

Cf. A058249.

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

A059959 Distance of 2^n from its nearest prime neighbor and in case of a tie, choose the smaller.

Original entry on oeis.org

-1, 0, 1, 1, -1, 1, 3, 1, -1, 3, 3, -5, 3, 1, 3, -3, -1, 1, -3, 1, 3, 9, 3, -9, 3, -35, 5, -29, -3, 3, -3, 1, 5, 9, -25, 31, 5, -9, -7, 7, -15, 21, 11, -29, -7, 55, -15, -5, -21, -69, 27, -21, -21, -5, 33, -3, 5, -9, 27, 55, -33, 1, 57, 25, -13, 49, 5, -3, 23, 19, -25, -11, -15, -29, 35, -33, 15, -11, -7, -23, -13, -17, -9, 55, -3, 19

Offset: 0

Labos Elemer, Mar 02 2001

sign

			n=19, 2^19=524288, prevprime(524288)=524287, nextprime(524288)=524309, so min{21,1}=1=a(19).

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (terms 0-4500 from Giovanni Resta)

Cf. A000079, A117387, A013597, A013603, A014210, A014234, A058249.

with(numtheory): [seq(min(nextprime(2^i)-2^i, 2^i-prevprime(2^i)), i=2..100)];

f[n_] := Block[{k = 0}, While[ !PrimeQ[2^n -k] && !PrimeQ[2^n +k], k++]; If[ PrimeQ[2^n -k], k, -k]]; Array[f, 70, 0] (* Robert G. Wilson v, Mar 14 2006 and modified Jan 12 2024 *)

a(n) = A000079(n) - A117387(n).

Signs added by Robert G. Wilson v, Mar 14 2006

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.

A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

Original entry on oeis.org

0, 1, -1, 2, 0, 1, -2, 3, 2, -2, 0, 8, 12, -8, -7, 14, -1, 10, 2, 4, 6, -3, 20, -2, 5, -5, -27, 4, -16, 5, 5, 4, -8, 11, 13, -8, -19, 8, -36, 3, 2, -14, -5, 2, -3, -55, -19, -6, 14, -54, -13, -53, 63, -26, 38, -2, 21, 38, -30, 7, 39, 2, -23, 41, 2, -8, 5, 5, -5, -110

Offset: 2

Hugo Pfoertner, Jan 29 2021

sign

a(n) > 0 if the difference nextprime(2^n) - 2^n = A013597(n) is greater than the difference 2^n - previousprime(2^n) = A013603(n).

			a(4) = -1: 2^4 = 16, (13 + 17 - 32)/2 = -1;
a(5) = 2: 2^5 = 32, (31 + 37 - 64)/2 = 2;
a(6) = 0: 2^6 = 64, (61 + 67 - 128)/2 = 0.

Hugo Pfoertner, Table of n, a(n) for n = 2..5000 (from T. D. Noe's b-files in A013597 and A013603).

Cf. A151799, A151800, A013597, A013603, A058249, A226178.

a:= (p-> (nextprime(p)+prevprime(p))/2-p)(2^n):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 29 2021

Array[(NextPrime[2^#] + NextPrime[2^#, -1] - 2^(# + 1))/2 &, 60, 2] (* Michael De Vlieger, Aug 07 2022 *)

for(k=2,71,my(p2=2^k,pp=precprime(p2),pn=nextprime(p2));if(print1((pp+pn-2*p2)/2", ")))

a(n) = (A013597(n) - A013603(n))/2.

a(A226178(n)) = 0.

Name made more precise by Peter Luschny, Aug 08 2022

A060271 Difference between smallest prime following and largest prime preceding 2*(n-th prime).

Original entry on oeis.org

2, 2, 4, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 6, 8, 4, 14, 14, 6, 10, 10, 6, 4, 6, 4, 12, 12, 12, 12, 4, 6, 6, 6, 4, 14, 14, 4, 14, 6, 10, 6, 8, 4, 6, 8, 4, 10, 6, 8, 4, 4, 12, 8, 4, 12, 18, 18, 6, 10, 6, 6, 10, 4, 12, 12, 10, 12, 4, 10, 10, 8, 10, 6, 8, 4, 8, 14, 10, 12, 10, 10, 14, 4, 14, 4, 4, 20, 8

Offset: 1

Labos Elemer, Mar 23 2001

nonn

			For n = 1: prime(1) = 2, 2*prime(1) = 4 is between 3 and 5, their difference is 2 = a(1).
For n = 6: prime(6) = 13, 2*prime(6) = 26 is between 23 and 29 and their difference is 6 = a(6).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A059786, A059787, A059788, A059789, A059790, A013633, A058043, A058249, A058044, A060267, A001223.

with(numtheory): [seq(nextprime(2*ithprime(j))-prevprime(2*ithprime(j)),j=1...256)];

dsplp[n_]:=Module[{np=2Prime[n]},NextPrime[np]-NextPrime[np,-1]]; Array[ dsplp,90] (* Harvey P. Dale, Mar 20 2013 *)

a(n) = {my(m = 2*prime(n)); nextprime(m+1) - precprime(m-1);} \\ Amiram Eldar, Feb 08 2025

Offset changed to 1 and a(1) prepended by Amiram Eldar, Feb 08 2025

A092507 (Smallest prime >= 2^n) + (largest prime <= 2^n).

Original entry on oeis.org

2, 4, 8, 18, 30, 68, 128, 258, 508, 1030, 2052, 4092, 8192, 16400, 32792, 65520, 131058, 262172, 524286, 1048596, 2097156, 4194312, 8388620, 16777210, 33554472, 67108860, 134217738, 268435446, 536870858, 1073741832, 2147483616

Offset: 0

Jorge Coveiro, Apr 05 2004

nonn

For n=0 we just take a(0)=2, the least prime >= 2^0, as there is no prime <= 2^0. - Robert Israel, Nov 01 2018

Robert Israel, Table of n, a(n) for n = 0..3317

Cf. A014210, A014234, A058249.

[2, seq( (nextprime(2^x-1)+prevprime(2^x+1)),x=1..20)]; # Corrected by Robert Israel, Nov 01 2018

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[PrevPrim[2^n+1] + NextPrim[2^n-1], {n, 31}] (* Robert G. Wilson v, Apr 14 2004 *)

a(n) = A014210(n) + A014234(n) for n >= 2. - Robert Israel, Nov 01 2018

More terms from Robert G. Wilson v, Apr 14 2004

cp's OEIS Frontend

A331620 Records of A058249: (Smallest prime >= 2^n) - (largest prime <= 2^n).

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

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A059959 Distance of 2^n from its nearest prime neighbor and in case of a tie, choose the smaller.

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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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A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

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A060271 Difference between smallest prime following and largest prime preceding 2*(n-th prime).

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