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User: Sam Vaseghi

Sam Vaseghi has authored 2 sequences.

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

Sam Vaseghi, Jun 02 2013

nonn

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.

			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,...

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

Cf. A166517, A226233.

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}]

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.

A226233 Ten copies of each positive integer.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10

Offset: 1

Sam Vaseghi, Jun 01 2013

Class of well and totally ordered sequences of (p-1)-tuples of natural numbers for p = 11.
Given a prime p the class of sequences a(n,p) can be constructed. The above example is for p=11. The class of well and totally ordered sequences of (prime-1)-tuples of natural numbers contains all sequences a(n) according to FORMULA for primes p. The class is crucial and will be applied to define other sequences, that will be submitted to OEIS as well a posterior.
a(n) = A132272(n-1) for n<=200, but the two sequences start to differ then. - R. J. Mathar, Jun 13 2025

S. Vaseghi (alias al-Hwarizmi), Combination of positive integers in terms of primes (sophisticated version 2)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A000027, A004526, A002265.

Cf. A059995 (10 copies of nonnegative integers).

A226233 := proc(n)
    option remember ;
    if n <= 10 then
        1;
    elif n <=20 then
        2;
    else
        procname(n-1)+procname(n-10)-procname(n-11) ;
    end if;
end proc:
seq(A226233(n),n=1..120) ; # R. J. Mathar, Jun 13 2025

p=11; k = (p - 1); alpha = (k + n - 1 - (Mod[(n - 1), k]))/k; Table[alpha, {n, 100}]
Table[PadRight[{},10,n],{n,10}]//Flatten (* Harvey P. Dale, May 24 2021 *)

a(n)=(n+9)\10 \\ Charles R Greathouse IV, Jun 05 2013

a(n,p) = ((p-1) + n - (1 + ((n-1) mod (p-1))))/(p-1); p is a prime and n positive integer; for this sequence p = 11.

G.f.: x / ( (1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Jun 13 2025

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User: Sam Vaseghi

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

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