

A106044


Difference between nth prime and next larger perfect square.


11



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
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OFFSET

1,1


COMMENTS

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 nth 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


LINKS

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


EXAMPLE

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)


MATHEMATICA

lst={}; Do[p=Prime[n]; s=p^(1/2); f=Floor[s]; a=(f+1)^2; d=ap; 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 *)


PROG

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


CROSSREFS

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.
Sequence in context: A054269 A086450 A270439 * A124896 A008742 A029136
Adjacent sequences: A106041 A106042 A106043 * A106045 A106046 A106047


KEYWORD

nonn,easy,tabf


AUTHOR

Zak Seidov, May 06 2005


EXTENSIONS

Edited by M. F. Hasler, Oct 19 2018


STATUS

approved



