A054271 -id:A054271 - cp's OEIS frontend

A091666 Difference between prime(n)^2 and the next prime.

1, 2, 4, 4, 6, 4, 4, 6, 12, 12, 6, 4, 12, 12, 4, 10, 10, 6, 4, 10, 4, 6, 10, 6, 4, 10, 4, 18, 6, 12, 10, 6, 4, 12, 28, 6, 10, 4, 4, 18, 10, 10, 12, 4, 12, 6, 10, 10, 10, 12, 4, 10, 18, 28, 18, 22, 6, 12, 4, 16, 18, 4, 4, 10, 4, 4, 6, 22, 4, 42, 24, 22, 10, 4

Offset: 1

Pierre CAMI, Jan 27 2004

Conjecturally, a(n) << log^2 n (with constant around 8/e^gamma in the supremum). [Charles R Greathouse IV, Dec 27 2011]

Except for a(2)=2, there are no terms = 2 mod 6 (as p^2+2 = 0 mod 3 for primes p > 3). Also, only 1 and 2 appear once while all other terms may appear (infinitely) many times. [Zak Seidov, Apr 18 2012]

			prime(3)=5, 5*5=25 for k=4 25+4=29 prime, k=4 is the least k with prime(3)^2 + k prime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A001248, A062772, A054271, A133517, A133518, A133519, A133520, A133521, A133522, A001223.

NextPrime[#^2]-#^2&/@Prime[Range[74]] (* Zak Seidov, Apr 18 2012 *)

a(n) = my(x=prime(n)^2); nextprime(x)-x; \\ Michel Marcus, Oct 07 2023

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} log(prime(n))) = 2. - Alain Rocchelli, Oct 04 2023

A054270 Largest prime below prime(n)^2 (A001248).

Original entry on oeis.org

3, 7, 23, 47, 113, 167, 283, 359, 523, 839, 953, 1367, 1669, 1847, 2207, 2803, 3469, 3719, 4483, 5039, 5323, 6229, 6883, 7919, 9403, 10193, 10607, 11447, 11867, 12763, 16127, 17159, 18757, 19319, 22193, 22787, 24631, 26561, 27883, 29927, 32029

Offset: 1

Labos Elemer, May 05 2000

nonn

For n > 1, the n-1 first primes determine the primes up to a(n). This is how the Sieve of Eratosthenes works. - Jean-Christophe Hervé, Oct 21 2013

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000879, A001248, A054271.

seq(prevprime(ithprime(i)^2),i=1..100); # Robert Israel, May 04 2020

NextPrime[Prime[Range[50]]^2,-1] (* Harvey P. Dale, May 19 2016 *)

a(n) = precprime(prime(n)^2); \\ Michel Marcus, Dec 13 2013

a(n) = Max[prime q: q < prime(n)^2].

a(n) = prime(A000879(n)) = A000040(A000879(n)). - Jean-Christophe Hervé, Oct 21 2013

A133517 Smallest k such that p(n)^3 - k is prime where p(n) is the n-th prime.

Original entry on oeis.org

1, 4, 12, 6, 4, 18, 4, 2, 4, 10, 2, 2, 4, 14, 10, 4, 22, 38, 2, 28, 14, 12, 4, 22, 24, 4, 14, 24, 2, 10, 14, 4, 16, 12, 10, 2, 12, 30, 10, 16, 48, 18, 10, 20, 30, 42, 2, 14, 4, 26, 18, 10, 2, 10, 4, 4, 16, 12, 2, 34, 24, 58, 30, 4, 38, 6, 14, 14, 10, 12, 36, 6, 2, 24, 68, 4, 6, 26, 10

Offset: 1

Carl R. White, Sep 14 2007

			p(4)=7, 7^3 = 343; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
for k = 2: 343 - 2 = 341, which is 11 * 31 and not prime.
for k = 4: 343 - 4 = 339, which is 3 * 113, also not prime.
for k = 6: 343 - 6 = 337, which is prime, so 6 is the smallest number that can be subtracted from 343 to make another prime.
Hence a(4) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030078, A054271, A091666, A133518, A133519, A133520, A133521, A133522, (A001223).

sk[n_]:=With[{c=Prime[n]^3},c-NextPrime[c,-1]]; Array[sk,80] (* Harvey P. Dale, May 07 2019 *)

a(n) = {k = 0; while (! isprime(prime(n)^3 - k), k++); return (k);} \\Michel Marcus, Aug 02 2013

A133518 Smallest k such that p(n)^3 + k is prime where p(n) is the n-th prime.

Original entry on oeis.org

3, 2, 2, 4, 30, 6, 6, 4, 30, 2, 12, 18, 6, 24, 14, 14, 12, 10, 16, 2, 6, 4, 2, 14, 54, 6, 4, 18, 4, 2, 30, 26, 56, 10, 24, 12, 24, 10, 30, 2, 18, 6, 26, 24, 14, 28, 18, 10, 14, 10, 12, 24, 16, 6, 18, 2, 20, 6, 4, 12, 4, 6, 10, 2, 6, 14, 16, 4, 18, 10, 14, 14, 16, 24, 4, 12, 32, 16, 50, 12, 2

Offset: 1

Carl R. White, Sep 14 2007

			p(1)=2, 2^3 = 8. for even k, 2^r + k is even and thus not prime, so we only need consider odd k.
for k = 1: 8 + 1 = 9, which is 3^2 and not prime.
for k = 3: 8 + 3 = 11, which is prime, so 3 is the smallest number that can be added to 8 to make a new prime.
Hence a(1) = 3.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A030078, A054271, A091666, A133517, A133519, A133520, A133521, A133522, (A001223).

[NextPrime(p^3)-p^3: p in PrimesUpTo(500)]; // Bruno Berselli, Sep 03 2013

Table[NextPrime[Prime[n]^3] - Prime[n]^3, {n, 100}] (* Bruno Berselli, Sep 03 2013 *)

a(n) = {k = 0; p3 = prime(n)^3; while (! isprime(p3+k), k++); k;} \\ Michel Marcus, Sep 03 2013

a(n) = {p3 = prime(n)^3; nextprime(p3) - p3;} \\ Michel Marcus, Sep 03 2013

A133519 Smallest k such that p(n)^4 - k is prime where p(n) is the n-th prime.

Original entry on oeis.org

3, 2, 6, 2, 2, 2, 24, 14, 18, 2, 8, 8, 2, 2, 12, 2, 2, 24, 24, 38, 2, 8, 2, 54, 12, 2, 12, 12, 44, 18, 14, 18, 12, 32, 12, 24, 38, 14, 12, 12, 54, 2, 50, 8, 32, 8, 12, 14, 24, 8, 8, 2, 2, 12, 18, 30, 50, 12, 2, 24, 12, 2, 32, 2, 84, 12, 8, 12, 8, 74, 14, 18, 2, 20, 24, 14, 2, 14, 14, 2, 18

Offset: 1

Carl R. White, Sep 14 2007

			p(3)=5, 5^4 = 625; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
for k = 2: 625 - 2 = 623, which is 7 * 89 and not prime.
for k = 4: 625 - 4 = 621, which is 3^3 * 23, also not prime.
for k = 6: 625 - 6 = 619, which is prime, so 6 is the smallest number that can be subtracted from 625 to make another prime.
Hence a(3) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030814, A054271, A091666, A133517, A133518, A133520, A133521, A133522, (A001223).

sk[p_]:=Module[{k=1,c=p^4},While[CompositeQ[c-k],k++];k]; sk/@Prime[Range[100]] (* Harvey P. Dale, Nov 19 2023 *)
Table[With[{c=p^4},c-NextPrime[c,-1]],{p,Prime[Range[100]]}] (* Harvey P. Dale, Nov 20 2023 *)

A133520 Smallest k such that p(n)^4 + k is prime where p(n) is the n-th prime.

Original entry on oeis.org

1, 2, 6, 10, 12, 10, 16, 16, 6, 12, 18, 16, 12, 28, 6, 22, 6, 16, 6, 16, 6, 16, 30, 6, 16, 42, 22, 42, 28, 52, 22, 16, 28, 10, 28, 70, 30, 42, 78, 36, 12, 42, 6, 12, 40, 12, 12, 16, 16, 16, 18, 10, 6, 22, 60, 46, 76, 46, 18, 126, 12, 22, 22, 6, 16, 16, 22, 18, 120, 22, 12, 6, 6, 36

Offset: 1

Carl R. White, Sep 14 2007

			p(2)=3, 3^4 = 81; for odd k and n > 1, p(n)^r + k is even and thus not prime, so we only need consider even k.
for k = 2: 81 + 2 = 83, which is prime, so 2 is the smallest number that can be added to 81 to make a new prime.
Hence a(2) = 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030814, A054271, A091666, A133517, A133518, A133519, A133521, A133522, (A001223).

NextPrime[#]-#&/@(Prime[Range[80]]^4) (* Harvey P. Dale, May 17 2015 *)

A133521 Smallest k such that p(n)^5 - k is prime where p(n) is the n-th prime.

Original entry on oeis.org

1, 2, 4, 20, 4, 2, 18, 18, 16, 6, 2, 6, 24, 12, 36, 22, 10, 8, 8, 24, 20, 86, 22, 6, 18, 42, 26, 6, 50, 52, 20, 12, 48, 2, 196, 68, 18, 14, 16, 16, 18, 2, 10, 6, 16, 38, 2, 36, 6, 2, 16, 42, 18, 42, 40, 34, 22, 2, 38, 4, 36, 52, 26, 132, 36, 28, 24, 74, 46, 36, 4, 16, 8, 24, 80, 16

Offset: 1

Carl R. White, Sep 14 2007

			p(10)=29, 29^5 = 20511149; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
for k = 2: 20511149 - 2 = 20511147, which is 3 * 23 * 297263 and not prime.
for k = 4: 20511149 - 4 = 20511145, which is 5 * 4102229, also not prime.
for k = 6: 20511149 - 6 = 20511141, which is prime, so 6 is the smallest number that can be subtracted from 20511149 to make another prime.
Hence a(10) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050997, A054271, A091666, A133517, A133518, A133519, A133520, A133522, (A001223).

#-NextPrime[#,-1]&/@(Prime[Range[80]]^5) (* Harvey P. Dale, Sep 27 2020 *)

A133522 Smallest k such that p(n)^5 + k is prime where p(n) is the n-th prime.

Original entry on oeis.org

5, 8, 12, 4, 2, 6, 20, 22, 8, 8, 12, 22, 26, 30, 20, 20, 74, 52, 22, 26, 4, 22, 6, 42, 40, 8, 58, 44, 42, 8, 40, 6, 36, 28, 2, 28, 6, 4, 20, 14, 2, 12, 8, 46, 2, 40, 10, 4, 110, 12, 18, 44, 42, 6, 24, 20, 8, 28, 46, 2, 18, 6, 60, 36, 24, 2, 18, 4, 24, 48, 6, 30, 6, 6, 22, 6, 2, 6, 2, 40, 2

Offset: 1

Carl R. White, Sep 14 2007

			p(2)=3, 3^5 = 243; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
for k = 2: 243 + 2 = 245, which is 5 * 7^2 and not prime.
for k = 4: 243 + 4 = 247, which is 13 * 19, also not prime.
for k = 6: 243 + 6 = 249, which is 3 * 83, also not prime.
for k = 8: 243 + 8 = 251, which is prime, so 8 is the smallest number that can be added to 243 to make another prime.
Hence a(2) = 8.

Cf. A050997, A054271, A091666, A133517, A133518, A133519, A133520, A133521, (A001223).

A062772 Smallest prime larger than square of n-th prime.

Original entry on oeis.org

5, 11, 29, 53, 127, 173, 293, 367, 541, 853, 967, 1373, 1693, 1861, 2213, 2819, 3491, 3727, 4493, 5051, 5333, 6247, 6899, 7927, 9413, 10211, 10613, 11467, 11887, 12781, 16139, 17167, 18773, 19333, 22229, 22807, 24659, 26573, 27893, 29947, 32051

Offset: 1

Labos Elemer, Jul 18 2001

Subsequence of A007491. - Zak Seidov, Apr 30 2015

			100th prime, 541 immediately follows 529, square of 9th prime.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A054270, A054271, A000879.

Cf. A007491. - Zak Seidov, Apr 30 2015

with(numtheory): [seq(nextprime(ithprime(w)^2),w=1..100)];

Array[NextPrime[Prime[#]^2] &, 41] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = { nextprime(prime(n)^2) } \\ Harry J. Smith, Aug 10 2009

a(n) = A007918(A001248(n)) = A151800(A001248(n)). - Michel Marcus, Jun 24 2014

a(n) = A007491(A000040(n)). - Zak Seidov, Apr 30 2015

A062773 Index of the smallest prime which follows square of n-th prime.

Original entry on oeis.org

3, 5, 10, 16, 31, 40, 62, 73, 100, 147, 163, 220, 264, 284, 330, 410, 488, 520, 610, 676, 706, 812, 887, 1001, 1164, 1253, 1295, 1382, 1424, 1524, 1878, 1977, 2142, 2191, 2490, 2548, 2730, 2916, 3044, 3242, 3437, 3513, 3869, 3946, 4090, 4165, 4628

Offset: 1

Labos Elemer, Jul 18 2001

nonn

			100th prime, 541 immediately follows 529, square of 9th prime, a(9)=100.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A054270, A054271, A000879.

PrimePi[NextPrime[#]]&/@(Prime[Range[50]]^2) (* Harvey P. Dale, Apr 12 2023 *)

a(n) = { primepi(prime(n)^2) + 1 } \\ Harry J. Smith, Aug 10 2009

a(n) = pi( nextprime( prime(n)^2 ) ).

a(n) = A000720(A062772(n)). - Michel Marcus, Jun 24 2014

Offset changed from 0 to 1 by Harry J. Smith, Aug 10 2009

cp's OEIS Frontend

A091666 Difference between prime(n)^2 and the next prime.

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A054270 Largest prime below prime(n)^2 (A001248).

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A133517 Smallest k such that p(n)^3 - k is prime where p(n) is the n-th prime.

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A133518 Smallest k such that p(n)^3 + k is prime where p(n) is the n-th prime.

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Magma

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A133519 Smallest k such that p(n)^4 - k is prime where p(n) is the n-th prime.

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A133520 Smallest k such that p(n)^4 + k is prime where p(n) is the n-th prime.

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A133521 Smallest k such that p(n)^5 - k is prime where p(n) is the n-th prime.

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A133522 Smallest k such that p(n)^5 + k is prime where p(n) is the n-th prime.

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