A059865
Product_{i=4..n} (prime(i) - 6).
Original entry on oeis.org
1, 1, 1, 1, 5, 35, 385, 5005, 85085, 1956955, 48923875, 1516640125, 53082404375, 1964048961875, 80526007436875, 3784722349533125, 200590284525255625, 11032465648889059375, 672980404582232621875, 43743726297845120421875
Offset: 1
a(7) = (prime(4)-6) * (prime(5)-6) * (prime(6)-6) * (prime(7)-6) = 1 * 5* 7 *11 = 385
Also in one period of dRRS with 2,6,30,210,2310,... modulus [A002110(n)] 1,2,8,48,480,... differences occur [A005867(n)]. The number of X42424Y residue-difference-patterns are 0,1,1,1,5,... respectively starting at suitable residues coprime to A002110(n).
- See A059862 for references.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
A059863
a(n) = Product_{i=3..n} (prime(i)-4).
Original entry on oeis.org
1, 1, 1, 3, 21, 189, 2457, 36855, 700245, 17506125, 472665375, 15597957375, 577124422875, 22507852492125, 967837657161375, 47424045200907375, 2608322486049905625, 148674381704844620625, 9366486047405211099375, 627554565176149143658125, 43301264997154290912410625
Offset: 1
- See A059862 for references.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
A059864
a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.
Original entry on oeis.org
1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568
Offset: 1
- 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
- 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.
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[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023
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Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)
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a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017
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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
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