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
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.
Crossrefs
Partial sums of A076821. - Michel Marcus, May 26 2018
Cf. A001223.
Programs
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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:
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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 *) -
PARI
a(n) = sum(k=1, n, (prime(k+1) - prime(k))^2); \\ Michel Marcus, May 26 2018
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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
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)A000720(n).