A076821 Squares of the differences between consecutive primes.

1, 4, 4, 16, 4, 16, 4, 16, 36, 4, 36, 16, 4, 16, 36, 36, 4, 36, 16, 4, 36, 16, 36, 64, 16, 4, 16, 4, 16, 196, 16, 36, 4, 100, 4, 36, 36, 16, 36, 36, 4, 100, 4, 16, 4, 144, 144, 16, 4, 16, 36, 4, 100, 36, 36, 36, 4, 36, 16, 4, 100, 196, 16, 4, 16, 196, 36, 100, 4, 16, 36, 64, 36

Offset: 1

Author

Cino Hilliard, Nov 19 2002

Keywords

• easy
• nonn

Comments

The sum of reciprocals is likely divergent, especially if the twin-prime conjecture is true.

The sum of the reciprocals diverges. In particular, the sum of the reciprocals up to n is at least n/(log n + log log n)^2 for n > 3. - Charles R Greathouse IV, Apr 17 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• B. Apostol, L. Panaitopol, L Petrescu, L. Toth, Some Properties of a Sequence Defined with the Aid of Prime Numbers, J. Int. Seq. 18 (2015) # 15.5.5.

Programs

Magma

[(NthPrime(n+1)-NthPrime(n))^2: n in [1..80]]; // Vincenzo Librandi, Jun 08 2016

Mathematica

Table[(Prime[n + 1] - Prime[n])^2, {n, 80}] (* Vincenzo Librandi, Jun 08 2016 *)

PARI

a(n)=(prime(n+1)-prime(n))^2 \\ Charles R Greathouse IV, Apr 17 2012

Extensions

Edited by Don Reble, May 03 2006