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%I #27 Apr 03 2023 10:36:10
%S 1,1,1,2,12,96,1152,16128,290304,6967296,181149696,5796790272,
%T 208684449792,7930009092096,333060381868032,15986898329665536,
%U 863292509801938944,48344380548908580864,2997351594032332013568
%N a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.
%C Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
%D R. K. Guy, Unsolved Problems in Number Theory, A8, A1
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
%D G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954
%H G. C. Greubel, <a href="/A059864/b059864.txt">Table of n, a(n) for n = 1..350</a>
%H C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=PrimeKtupleConjecture">Prime k-tuple Conjecture</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/hrdyltl/hrdyltl.html">Hardy-Littlewood Constants </a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010614100031/http://www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html">Hardy-Littlewood Constants </a> [From the Wayback machine]
%H G. H. Hardy and J. E. Littlewood, <a href="http://dx.doi.org/10.1007/BF02403921">Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%H G. Polya, <a href="http://www.jstor.org/stable/2308748">Heuristic reasoning in the theory of numbers</a>, Am. Math. Monthly,66 (1959), 375-384.
%t Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* _Harvey P. Dale_, Dec 29 2018 *)
%o (PARI) a(n) = prod(k=4, n, prime(k)-5); \\ _Michel Marcus_, Dec 12 2017
%o (Magma) [n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // _G. C. Greubel_, Feb 02 2023
%o (SageMath)
%o def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
%o [A059864(n) for n in range(1,31)] # _G. C. Greubel_, Feb 02 2023
%Y Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
%Y Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
%Y Cf. A059864, A059865.
%K nonn
%O 1,4
%A _Labos Elemer_, Feb 28 2001
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