A059865 Product_{i=4..n} (prime(i) - 6).

1, 1, 1, 1, 5, 35, 385, 5005, 85085, 1956955, 48923875, 1516640125, 53082404375, 1964048961875, 80526007436875, 3784722349533125, 200590284525255625, 11032465648889059375, 672980404582232621875, 43743726297845120421875

Offset: 1

Author

Labos Elemer, Feb 28 2001

Keywords

• nonn

Comments

Arises in Hardy-Littlewood prime k-tuplet conjectural formulas. Also the sequence gives the exact numbers of X42424Y difference-pattern in dRRS[m], where m=modulus=A002110(n). See A049296 (=dRRS[210]=list of first differences of reduced residue system modulo 210=4th primorial). A pattern X42424Y corresponds to a residue-sextuple or it is their difference-quintuple, X,Y > 4. Analogous pattern for primes is in A022008.

a(352) has 1001 decimal digits. - Michael De Vlieger, Mar 06 2017

Examples

			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).
		

References

• See A059862 for references.
• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..351
• 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. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Crossrefs

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

Programs

Mathematica

Table[Product[Prime@ i - 6, {i, 4, n}], {n, 19}] (* Michael De Vlieger, Mar 06 2017 *)

PARI

a(n) = prod(k=4, n, prime(k) - 6); \\ Michel Marcus, Mar 06 2017