A059865 - OEIS

%I #27 Apr 03 2023 10:36:10

%S 1,1,1,1,5,35,385,5005,85085,1956955,48923875,1516640125,53082404375,

%T 1964048961875,80526007436875,3784722349533125,200590284525255625,

%U 11032465648889059375,672980404582232621875,43743726297845120421875

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

%C 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.

%C a(352) has 1001 decimal digits. - _Michael De Vlieger_, Mar 06 2017

%D See A059862 for references.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

%H Michael De Vlieger, <a href="/A059865/b059865.txt">Table of n, a(n) for n = 1..351</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. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]

%e a(7) = (prime(4)-6) * (prime(5)-6) * (prime(6)-6) * (prime(7)-6) = 1 * 5* 7 *11 = 385

%e  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).

%t Table[Product[Prime@ i - 6, {i, 4, n}], {n, 19}] (* _Michael De Vlieger_, Mar 06 2017 *)

%o (PARI) a(n) = prod(k=4, n, prime(k) - 6); \\ _Michel Marcus_, Mar 06 2017

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

%K nonn

%O 1,5

%A _Labos Elemer_, Feb 28 2001