A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

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

G. C. Greubel, Table of n, a(n) for n = 1..350

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. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

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

Crossrefs

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.

Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.

Cf. A059864, A059865.

Sequence in context: A219119 A052611 A340938 * A095338 A308820 A322717

Adjacent sequences: A059861 A059862 A059863 * A059865 A059866 A059867

Keyword

nonn

Author

Labos Elemer, Feb 28 2001

Status

approved