A056892 a(n) = square excess of the n-th prime.

1, 2, 1, 3, 2, 4, 1, 3, 7, 4, 6, 1, 5, 7, 11, 4, 10, 12, 3, 7, 9, 15, 2, 8, 16, 1, 3, 7, 9, 13, 6, 10, 16, 18, 5, 7, 13, 19, 23, 4, 10, 12, 22, 24, 1, 3, 15, 27, 2, 4, 8, 14, 16, 26, 1, 7, 13, 15, 21, 25, 27, 4, 18, 22, 24, 28, 7, 13, 23, 25, 29, 35, 6, 12, 18, 22, 28, 36, 1, 9, 19, 21

Offset: 1

Author

Henry Bottomley, Jul 05 2000

Keywords

• nonn

Examples

			a(5) = 2 since the 5th prime is 11 = 3^2 + 2.
From _M. F. Hasler_, Oct 19 2018: (Start)
Written as a table, starting a new row when a square is reached, the sequence reads:
  1, 2,    // = 2 - 1, 3 - 1 = {primes between 1^2 = 1 and 2^2 = 4} - 1
  1, 3,     // = 5 - 4, 7 - 4 = {primes between 2^2 = 4 and 3^2 = 9} - 4
  2, 4,      // = 11 - 9, 13 - 9 = {primes between 3^2 = 9 and 4^2 = 16} - 9
  1, 3, 7,    // = 17 - 16, 19 - 16, 23 - 16 = {primes between 16 and 25} - 16
  4, 6,        // = 29 - 25, 31 - 25 = {primes between 5^2 = 25 and 6^2 = 36} - 25
  1, 5, 7, 11,  // = {37, 41, 43, 47: primes between 6^2 = 36 and 7^2 = 49} - 36
  4, 10, 12,    // = {53, 59, 61: primes between 7^2 = 49 and 8^2 = 64} - 49
  3, 7, 9, 15,  // = {67, 71, 73, 79: primes between 8^2 = 64 and 9^2 = 81} - 64
  2, 8, 16,     // = {83, 89, 97: primes between 9^2 = 81 and 10^2 = 100} - 81
  etc. (End)
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000196, A002496, A048760, A053186, A056893 .. A056898.

When written as a table, row lengths are A014085, and row sums are A108314 - A014085 * A000290 = A320688.

Programs

Mathematica

lst={};Do[p=Prime[n];s=p^(1/2);f=Floor[s];a=f^2;d=p-a;AppendTo[lst,d],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
#-Floor[Sqrt[#]]^2&/@Prime[Range[90]] (* Harvey P. Dale, Jul 06 2014 *)

PARI

A056892(n)={my(p=prime(n));p-sqrtint(p)^2} \\ M. F. Hasler, Oct 04 2009

Formula

a(n) = A053186(A000040(n)).

a(n) = A000040(n) - A000006(n)^2. - M. F. Hasler, Oct 04 2009