A262247 Number of squares formed from a square composed of p^2 unit squares where p is n-th prime.

5, 14, 55, 140, 506, 819, 1785, 2470, 4324, 8555, 10416, 17575, 23821, 27434, 35720, 51039, 70210, 77531, 102510, 121836, 132349, 167480, 194054, 238965, 308945, 348551, 369564, 414090, 437635, 487369, 690880, 757966, 866525, 904890, 1113775, 1159076

Offset

1,1

Comments

Inspired by geometric interpretation of A001248.

Obviously, a(n) is a prime number, only for n = 1.

Subsequence of A000330.

Obviously, there is only one solution for a and b to the equation p*p = a*b where p is prime, if a > 1 and b > 1. So there can be only one rectangle that is composed of p^2 unit squares, if a > 1 and b > 1. That rectangle is the p X p square. Its uniqueness is a motivation for this sequence.

Links

Table of n, a(n) for n=1..36.

Formula

a(n) = prime(n) * (prime(n)+1) * (2*prime(n)+1) / 6.

a(n) = A000330(A000040(n)).

Example

For a square composed of 4 unit squares, there are 5 squares.

Mathematica

Table[Prime[n] (Prime[n] + 1) (2 Prime[n] + 1)/6, {n, 50}] (* Vincenzo Librandi, Sep 17 2015 *)
(#(#+1)(2#+1))/6&/@Prime[Range[40]] (* Harvey P. Dale, Sep 05 2022 *)

Prog

(PARI) a(n) = my(p=prime(n)); p * (p+1) * (2*p+1) / 6;
vector(40, n, a(n))
(Magma) [NthPrime(n)*(NthPrime(n)+1)*(2*NthPrime(n)+1)/6: n in [1..40]]; // Vincenzo Librandi, Sep 17 2015

Crossrefs

Cf. A000330, A001248, A000217, A000537.

Sequence in context: A055488 A177049 A127922 * A279511 A281698 A362395

Adjacent sequences: A262244 A262245 A262246 * A262248 A262249 A262250

Keyword

nonn,easy

Author

Altug Alkan, Sep 16 2015

Status

approved