%I #27 Aug 07 2013 15:07:19

%S 6,5,4,3,2,1,13,12,11,10,9,8,20,19,18,17,16,15,27,26,25,24,23,22,34,

%T 33,32,31,30,29,41,40,39,38,37,36,48,47,46,45,44,43,55,54,53,52,51,50,

%U 62,61,60,59,58,57,69,68,67,66,65,64,76,75,74,73,72,71,83

%N Class of sequences of (p-1)-tuples of reverse order of natural numbers for p = 7.

%C Given a prime p, the class of sequences a(n,p) can be constructed from linear combination of the two sequences b(n,p) (A010885) and c(n,p) (A226233), according to a(n,p) = c(n,p)*p - b(n,p) (see Formula below) that ensures uniqueness of the form q = a(n,p)*p^m according to the decomposition theorem Vaseghi 2013 (see link and reference below), for p prime, q a positive integer and m a positive integer or zero. The above example is for p=7. The class is crucial and will be applied to define other number theoretic sequences, that will be submitted to OEIS as well a posterior.

%H S. Vaseghi (alias al-Hwarizmi), <a href="http://math.stackexchange.com/questions/407890/combination-of-positive-integers-in-terms-of-primes-sophisticated-version-2">Combination of positive integers in terms of primes (sophisticated version 2)</a>

%F a(n,p) = c(n,p)*p - b(n,p), where b(n,p) = (1+[(n-1)mod(p-1)]) (see A010885) and c(n,p) = ((p-1)+n-(1+[(n-1)mod(p-1)]))/(p-1) (see A226233), with p = 7.

%e for p=2: 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,...

%e for p=3: 2,1,5,4,8,7,11,10,14,13,17,16,20,19,23,22,26,25,29,28,...

%e for p=5: 4,3,2,1,9,8,7,6,14,13,12,11,19,18,17,16,24,23,22,21,...

%e for p=7: 6,5,4,3,2,1,13,12,11,10,9,8,20,19,18,17,16,15,27,26,...

%t p = 7; k = p - 1; c = (k + n - 1 - Mod[n - 1, k])/k; b = 1 + Mod[n - 1, k]; Table[c*p - b, {n, 68}]

%Y Cf. A166517, A226233.

%K nonn

%O 1,1

%A _Sam Vaseghi_, Jun 02 2013