A074741 Sum of squares of gaps between consecutive primes.

1, 5, 9, 25, 29, 45, 49, 65, 101, 105, 141, 157, 161, 177, 213, 249, 253, 289, 305, 309, 345, 361, 397, 461, 477, 481, 497, 501, 517, 713, 729, 765, 769, 869, 873, 909, 945, 961, 997, 1033, 1037, 1137, 1141, 1157, 1161, 1305, 1449, 1465, 1469, 1485, 1521

Offset

1,2

References

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York, Heidelberg, 1994, problem A8.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Paul Erdös, Some problems on the distribution of prime numbers, C.I.M.E., Teoria dei numeri (1955).

D. R. Heath-Brown, Gaps between primes and the pair correlation of zeros of the zeta function, Acta Arithmetica, vol. XLI (1982), 85.

M. Wolf, Some conjectures on the gaps between consecutive primes, preprint IFTUWr 894//95, submitted to Asterisque.

Formula

a(n) = Sum_{k=1..n} (prime(k+1)-prime(k))^2 = Sum_{k=1..n} A001223(k)^2.

Asymptotic expressions: D. R. Heath-Brown's conjecture: Sum_{prime(n)<=N} (prime(n)-prime(n-1))^2 ~ 2*N*log(N). Marek Wolf's conjecture: Sum_{prime(n)<N} (prime(n)-prime(n-1))^2 = 2*N^2/pi(N) + error term(N), pi(N)=A000720(n).

Maple

with(numtheory): a := proc(n) option remember: if (n=1) then RETURN(1) else RETURN(a(n-1)+(ithprime(n+1)-ithprime(n))^2) fi: end:

Mathematica

Rest[FoldList[Plus, 0, (#[[2]] - #[[1]])^2 & /@ Partition[Prime[Range[100]], 2, 1]]]
nn=60; With[{dsp=Differences[Prime[Range[nn+1]]]^2}, Table[Total[Take[ dsp, n]], {n, nn}]] (* Harvey P. Dale, Nov 30 2011 *)
Accumulate[Differences[Prime[Range[60]]]^2] (* Harvey P. Dale, May 08 2015 *)

Prog

(PARI) a(n) = sum(k=1, n, (prime(k+1) - prime(k))^2); \\ Michel Marcus, May 26 2018
(Python)
from sympy import nextprime
from itertools import islice, accumulate
def gen():
    p, q = 2, 3
    while True:
        r = (q - p) ** 2
        yield r
        p, q = q, nextprime(q)
print(list(accumulate(islice(gen(), 51)))) # Adrienne Leonardo, Dec 18 2024

Crossrefs

Partial sums of A076821. - Michel Marcus, May 26 2018

Cf. A001223.

Sequence in context: A162907 A165345 A177240 * A166701 A116520 A273561

Adjacent sequences: A074738 A074739 A074740 * A074742 A074743 A074744

Keyword

nonn,nice

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 05 2002

Status

approved