A104492 -id:A104492 - cp's OEIS frontend

A158038 Difference between n-th prime and next cube.

6, 5, 3, 1, 16, 14, 10, 8, 4, 35, 33, 27, 23, 21, 17, 11, 5, 3, 58, 54, 52, 46, 42, 36, 28, 24, 22, 18, 16, 12, 89, 85, 79, 77, 67, 65, 59, 53, 49, 43, 37, 35, 25, 23, 19, 17, 5, 120, 116, 114, 110, 104, 102, 92, 86, 80, 74, 72, 66, 62, 60, 50, 36, 32, 30, 26, 12, 6, 165, 163

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

Could be read as a table, since there are always several primes between two cubes. Whenever a(n+1) > a(n), the n-th prime is the largest one below a given cube and prime(n+1) is the smallest prime larger than that cube. For n > 1, these are also the indices where the parity of the terms changes. - M. F. Hasler, Oct 19 2018

			The first terms are: 8 - 2 = 6, 8 - 3 = 5, 8 - 5 = 3, 8 - 7 = 1, 27 - 11 = 16, ...
From _M. F. Hasler_, Oct 19 2018: (Start)
Starting a new row when going to the next cube, the sequence reads:
  6, 5, 3, 1,                      // = 8 - {primes between 1^3 = 1 and 2^3 = 8}
  16, 14, 10, 8, 4,                // = 27 - {primes between 2^3 = 8 and 3^3 = 27}
  35, 33, 27, 23, 21, 17, 11, 5, 3, // = 64 - {primes between 27 and 4^3 = 64}
  58, 54, 52, ..., 18, 16, 12,     // = 125 - {primes between 64 and 5^3 = 125}
  89, 85, 79, ..., 19, 17, 5,      // = 216 - {primes between 125 and 6^3 = 216}
  120, 116, 114, ..., 26, 12, 6,   // = 343 - {primes between 216 and 7^3 = 343}
  etc. (End)

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A106044, A104492, A056892, A158037

lst={};Do[p=Prime[n];s=p^(1/3);f=Floor[s];a=(f+1)^3;d=a-p;AppendTo[lst,d],{n,6!}];lst
nc[n_]:=(Floor[Surd[n,3]]+1)^3-n; Table[nc[n],{n,Prime[Range[70]]}] (* Harvey P. Dale, Jun 19 2014 *)

A158038(n)=(sqrtnint(n=prime(n),3)+1)^3-n \\ M. F. Hasler, Oct 19 2018

first(n) = my(res = vector(n), t = 0, c = 2, c3 = 8); forprime(p = 2, oo, t++; if(p > c3, c++; c3 = c^3); res[t] = c3 - p; if(t==n, return(res))) \\ David A. Corneth, Oct 19 2018

a(n) > 0. - David A. Corneth, Oct 19 2018

Edited by M. F. Hasler, Oct 19 2018

A158037 A106044 sorted and duplicates removed.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

All natural numbers except squares, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49, 8^2=64, ... are missing from the current sequence.

Cf. A106044, A104492, A056892

lst={};Do[p=Prime[n];s=p^(1/2);f=Floor[s];a=(f+1)^2;d=a-p;AppendTo[lst,d],{n,7!}];Take[Union[lst],5! ]
(Floor[Sqrt[#]]+1)^2-#&/@Prime[Range[500]]//Union (* Harvey P. Dale, Aug 02 2021 *)

A104493 Numbers n for which the cube excess of the n-th prime is prime.

Original entry on oeis.org

2, 5, 6, 8, 10, 19, 20, 23, 26, 28, 31, 48, 49, 50, 51, 52, 55, 56, 57, 59, 61, 65, 66, 99, 100, 105, 110, 112, 114, 117, 121, 125, 127, 170, 171, 173, 178, 184, 185, 186, 190, 192, 194, 196, 200, 201, 206, 208, 214, 270, 271, 272, 274, 277, 278, 279, 280, 282

Offset: 1

Jonathan Vos Post, Mar 19 2005

			99 is an element of this sequence because the 99th prime is 523, 523 - 8^3 = 523-512 = 11 and 11 is prime. 100 is in this sequence because the 100th prime is 541 and 541-8^3 = 29, which is prime.

Cf. A000040, A053186, A055400, A056892, A102821, A104492.

f[n_] := Block[{k = 1, p = Prime[n]}, While[k^3 < p, k++ ]; p - (k - 1)^3]; Select[ Range[ 284], PrimeQ[ f[ # ]] &] (* Robert G. Wilson v, Mar 19 2005 *)

n such that A055400(A000040(n)) is an element of A000040. n such that A104492(n) is prime.

More terms from Robert G. Wilson v, Mar 19 2005

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A158038 Difference between n-th prime and next cube.

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A158037 A106044 sorted and duplicates removed.

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A104493 Numbers n for which the cube excess of the n-th prime is prime.

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