A158038 -id:A158038 - cp's OEIS frontend

2, 1, 4, 2, 5, 3, 8, 6, 2, 7, 5, 12, 8, 6, 2, 11, 5, 3, 14, 10, 8, 2, 17, 11, 3, 20, 18, 14, 12, 8, 17, 13, 7, 5, 20, 18, 12, 6, 2, 23, 17, 15, 5, 3, 28, 26, 14, 2, 29, 27, 23, 17, 15, 5, 32, 26, 20, 18, 12, 8, 6, 31, 17, 13, 11, 7, 30, 24, 14, 12, 8, 2, 33, 27, 21, 17, 11, 3, 40, 32, 22

Offset: 1

Zak Seidov, May 06 2005

Can be read as a table, since there are always several primes between two squares, although this is the yet unproved Legendre's conjecture, cf. A014085. Whenever a(n+1) > a(n), the n-th prime is the largest one below a given square and prime(n+1) is the smallest prime larger than that square. For n > 1, these are also the indices where the parity of the terms changes. - M. F. Hasler, Oct 19 2018

			From _M. F. Hasler_, Oct 19 2018: (Start)
Written as a table, starting a new row when a square is reached, the sequence reads:
  2, 1,  // 4 - {2, 3: primes between 1^2 = 1 and 2^2 = 4}
  4, 2,   // 9 - {5, 7: primes between 2^2 = 4 and 3^2 = 9}
  5, 3,    // 16 - {11, 13: primes between 3^2 = 9 and 4^2 = 16}
  8, 6, 2,  // 25 - {17, 19, 23: primes between 4^2 = 16 and 5^2 = 25}
  7, 5,      // 36 - {29, 31: primes between 5^2 = 25 and 6^2 = 36}
  12, 8, 6, 2,// 49 - {37, 41, 43, 47: primes between 6^2 = 36 and 7^2 = 49}
  11, 5, 3,    // 64 - {53, 59, 61: primes between 7^2 = 49 and 8^2 = 64}
  14, 10, 8, 2, // 81 - {67, 71, 73, 79: primes between 8^2 = 64 and 9^2 = 81}
  17, 11, 3,     // 100 - {83, 89, 97: primes between 9^2 = 81 and 10^2 = 100}
  etc. (End)

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

Cf. A158038 (analog for cubes).
Read as a table, row lengths are A014085 (number of primes between squares).
Row sums are A014085 * A000290(.+1) - A108314.

lst={};Do[p=Prime[n];s=p^(1/2);f=Floor[s];a=(f+1)^2;d=a-p;AppendTo[lst,d],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
(Floor[Sqrt[#]]+1)^2-#&/@Prime[Range[90]] (* Harvey P. Dale, Feb 08 2013 *)

A106044(n)=(sqrtint(n=prime(n))+1)^2-n \\ M. F. Hasler, Oct 19 2018

Edited by M. F. Hasler, Oct 19 2018

cp's OEIS Frontend

A106044 Difference between n-th prime and next larger perfect square.

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