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

Author

Labos Elemer, Feb 28 2001

Keywords

• nonn

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.

Crossrefs

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

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

Cf. A059864, A059865.

Programs

Magma

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Mathematica

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

PARI

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

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