A060199 -id:A060199 - cp's OEIS frontend

A121842 Difference between n^3 and next prime.

2, 1, 3, 2, 3, 2, 7, 4, 9, 4, 9, 30, 5, 6, 5, 14, 3, 6, 7, 4, 9, 16, 3, 30, 5, 4, 3, 4, 9, 2, 11, 12, 3, 14, 9, 24, 7, 18, 5, 14, 7, 6, 5, 24, 9, 2, 31, 14, 5, 10, 3, 10, 3, 14, 13, 18, 5, 28, 9, 12, 23, 10, 3, 2, 3, 2, 5, 16, 9, 2, 19, 2, 25, 6, 3, 16, 3, 6, 5, 4, 9, 16, 13, 2, 19, 4, 3, 4, 9, 14

Offset: 0

Zak Seidov, Aug 29 2006

From Ingham (1937) it follows that there is a prime between x^3 and (x+1)^3 if x is sufficiently large: see A060199 for further details. - M. F. Hasler, Nov 09 2020

			a(6)=7 because next prime after 6^3=216 is 223 and 223-216=7.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
A. E. Ingham, On the difference between consecutive primes. Quarterly Journal of Mathematics. Oxford Series. 8 (1) (1937) p. 255-266 (doi:10.1093/qmath/os-8.1.255).

Cf. A060199 (number of primes between consecutive cubes).

Array[NextPrime[#] - # &[#^3] &, 90, 0] (* Michael De Vlieger, Nov 12 2020 *)

a(n) = nextprime(n^3) - n^3; \\ Michel Marcus, Oct 10 2013

a(n) = A013632(n^3) = A013632(A000578(n)). - Michel Marcus, Oct 10 2013

A157481 Number of primes between n^3-n^2 and (n+1)^3-(n+1)^2.

Original entry on oeis.org

0, 2, 5, 8, 10, 16, 21, 24, 32, 36, 43, 53, 57, 65, 74, 86, 92, 104, 114, 123, 133, 150, 151, 175, 180, 194, 207, 224, 238, 251, 271, 275, 306, 305, 332, 349, 359, 383, 408, 410, 434, 458, 473, 497, 502, 549, 553, 570, 590, 630, 641, 668, 685, 718, 726, 748, 780

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Cf. A049451, A060199, A014085, A144391

Table[PrimePi[(n+1)^3-(n+1)^2]-PrimePi[n^3-n^2],{n,0,5!}]

A276119 Number of twin prime pairs between n^3 and (n+1)^3.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 4, 6, 5, 11, 9, 12, 11, 12, 17, 17, 16, 19, 16, 18, 24, 22, 17, 22, 26, 32, 36, 33, 26, 35, 39, 45, 36, 36, 38, 52, 42, 51, 40, 48, 55, 51, 67, 62, 64, 66, 66, 72, 77, 67, 71, 73, 96, 75, 69, 109, 83, 90, 86, 100, 101, 95, 91, 112, 111

Offset: 1

G. L. Honaker, Jr., Aug 21 2016

nonn

Is there a twin prime pair between all consecutive cubes?

			a(9)=5 because there are 5 twin prime pairs between 9^3 and 10^3, i.e., {809, 811}, {821, 823}, {827, 829}, {857, 859}, {881, 883}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000578, A001359, A006512, A060199.

N:= 100: # to get a(1) .. a(N)
Primes:= select(isprime, {seq(x,x=3..(N+1)^3,2)}):
Tprimes:= Primes intersect map(t -> t-2,Primes):
seq(nops(select(t -> t > n^3 and t < (n+1)^3-2, Tprimes)),n=1..N); # Robert Israel, Aug 21 2016

a(n)=my(p=nextprime(n^3),s); forprime(q=p+1, (n+1)^3, if(q-p==2, s++); p=q); s \\ Charles R Greathouse IV, Aug 21 2016

use ntheory ":all"; sub a276119 { my $n = shift; twin_prime_count($n**3,($n+1)**3); } # Dana Jacobsen, Aug 22 2016

A157482 Number of primes between n^3-n^2-n^1 and (n+1)^3-(n+1)^2-(n+1)^1.

Original entry on oeis.org

0, 1, 5, 8, 10, 16, 21, 24, 30, 39, 42, 52, 57, 65, 75, 86, 92, 102, 115, 122, 133, 150, 151, 176, 181, 192, 209, 221, 239, 252, 270, 273, 307, 308, 328, 350, 359, 383, 407, 414, 430, 460, 472, 494, 504, 548, 554, 571, 590, 629, 642, 669, 681, 722, 724, 749, 776

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Cf. A049451, A060199, A014085, A144391, A157481

Table[PrimePi[(n+1)^3-(n+1)^2-(n+1)^1]-PrimePi[n^3-n^2-n^1],{n,0,5!}]

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A121842 Difference between n^3 and next prime.

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A157481 Number of primes between n^3-n^2 and (n+1)^3-(n+1)^2.

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A276119 Number of twin prime pairs between n^3 and (n+1)^3.

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A157482 Number of primes between n^3-n^2-n^1 and (n+1)^3-(n+1)^2-(n+1)^1.

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