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A059864
a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.
1
1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568
OFFSET
1,4
COMMENTS
Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954
LINKS
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.
MATHEMATICA
Join[{1, 1, 1}, FoldList[Times, Prime[Range[4, 20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)
PROG
(PARI) a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017
(Magma) [n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023
(SageMath)
def A059864(n): return product(nth_prime(j) -5 for j in range(4, n+1))
[A059864(n) for n in range(1, 31)] # G. C. Greubel, Feb 02 2023
KEYWORD
nonn
AUTHOR
Labos Elemer, Feb 28 2001
STATUS
approved