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A074741
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Sum of squares of gaps between consecutive primes.
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3
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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
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OFFSET
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1,2
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REFERENCES
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P. Erdös, Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955).
R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York, Heidelberg, 1994, problem A8.
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LINKS
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FORMULA
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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).
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MAPLE
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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
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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 *)
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PROG
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(PARI) a(n) = sum(k=1, n, (prime(k+1) - prime(k))^2); \\ Michel Marcus, May 26 2018
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 05 2002
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STATUS
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approved
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