A226293 Class of sequences of (p-1)-tuples of reverse order of natural numbers for p = 7.

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, 33, 32, 31, 30, 29, 41, 40, 39, 38, 37, 36, 48, 47, 46, 45, 44, 43, 55, 54, 53, 52, 51, 50, 62, 61, 60, 59, 58, 57, 69, 68, 67, 66, 65, 64, 76, 75, 74, 73, 72, 71, 83

Offset: 1

Author

Sam Vaseghi, Jun 02 2013

Keywords

• nonn

Comments

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.

Examples

			for p=2: 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,...
for p=3: 2,1,5,4,8,7,11,10,14,13,17,16,20,19,23,22,26,25,29,28,...
for p=5: 4,3,2,1,9,8,7,6,14,13,12,11,19,18,17,16,24,23,22,21,...
for p=7: 6,5,4,3,2,1,13,12,11,10,9,8,20,19,18,17,16,15,27,26,...
		

Links

• S. Vaseghi (alias al-Hwarizmi), Combination of positive integers in terms of primes (sophisticated version 2)

Crossrefs

Cf. A166517, A226233.

Programs

Mathematica

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}]

Formula

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.