Search: seq:1,2,3,1,4
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A235791
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Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k copies of every positive integer in nondecreasing order, and the first element of column k is in row k(k+1)/2.
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+30
241
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1, 2, 3, 1, 4, 1, 5, 2, 6, 2, 1, 7, 3, 1, 8, 3, 1, 9, 4, 2, 10, 4, 2, 1, 11, 5, 2, 1, 12, 5, 3, 1, 13, 6, 3, 1, 14, 6, 3, 2, 15, 7, 4, 2, 1, 16, 7, 4, 2, 1, 17, 8, 4, 2, 1, 18, 8, 5, 3, 1, 19, 9, 5, 3, 1, 20, 9, 5, 3, 2, 21, 10, 6, 3, 2, 1, 22, 10, 6, 4, 2, 1, 23, 11, 6, 4, 2, 1, 24, 11, 7, 4, 2, 1
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OFFSET
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1,2
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COMMENTS
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The alternating sum of the squares of the elements of the n-th row equals the sum of all divisors of all positive integers <= n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*(T(n,k))^2 = A024916(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
The sum of row n gives A060831(n), the sum of the number of odd divisors of all positive integers <= n. - Omar E. Pol, Mar 01 2014. [An equivalent assertion is that the sum of row n of A237048 is the number of odd divisors of n, and this was proved by Hartmut F. W. Hoft in a comment in A237048. - N. J. A. Sloane, Dec 07 2020]
The place to start is with A235791, which is very simple. Then go to A237591, also very simple, and A237593, still very simple.
You then need to interpret the rows of A237593 as Dyck paths. This interpretation is in terms of run lengths, so 2,1,1,2 means up twice, down once, up once, and down twice. Because the rows of A237593 are symmetric and of even length, this path will always be symmetric.
Now the surprising fact is that the areas enclosed by the Dyck path for n (laid on its side) always includes the area enclosed for n-1; and the number of squares added is sigma(n).
Finally, look at the connected areas enclosed by n but not by n-1; the size of these areas is the symmetric representation of sigma. (End)
Mathematica function has been written to check the first property up to n = 20000.
T(n,(sqrt(8n+1)-1)/2+1) = 0 for all n >= 1, which is useful for formulas for A237591 and A237593. (End)
Conjecture: T(n,k) is also the total number of partitions of all positive integers <= n into exactly k consecutive parts, i.e., the partial column sum of A285898, or in accordance with the triangles of the same family: the partial column sum of A237048. - Omar E. Pol, Apr 28 2017, Nov 24 2020
The above conjecture is true. The proof will be added soon (it uses the generating function for the columns). - N. J. A. Sloane, Nov 24 2020
T(n,k) is also the total length of all line segments between the k-th vertex and the central vertex of the largest Dyck path of the symmetric representation of sigma(n). In other words: T(n,k) is the sum of the last (A003056(n)-k+1) terms of the n-th row of A237591. - Omar E. Pol, Sep 07 2021
T(n,k) is also the Manhattan distance between the k-th vertex and the central vertex of the Dyck path described in the n-th row of the triangle A237593. - Omar E. Pol, Jan 11 2023
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LINKS
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FORMULA
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T(n,k) = ceiling((n+1)/k - (k+1)/2) for 1 <= n, 1 <= k <= floor((sqrt(8n+1)-1)/2) = A003056(n). - Hartmut F. W. Hoft, Apr 07 2014
G.f. for column k (k >= 1): x^(k*(k+1)/2)/( (1-x)*(1-x^k) ). - N. J. A. Sloane, Nov 24 2020
Sigma(n) = Sum_{k=1..A003056(n)} (-1)^(k-1) * (T(n,k)^2 - T(n-1,k)^2), assuming that T(k*(k+1)/2-1,k) = 0. - Omar E. Pol, Oct 10 2018
a(s(n,k)) = T(n,k), n >= 1, 1 <= k <= r = floor((sqrt(8*n + 1) - 1)/2), where s(n,k) = r*n - r*(r+1)*(r+2)/6 + k translates position (row n, column k) in the triangle of this sequence to its position in the sequence. - Hartmut F. W. Hoft, Feb 24 2021
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EXAMPLE
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Triangle begins:
1;
2;
3, 1;
4, 1;
5, 2;
6, 2, 1;
7, 3, 1;
8, 3, 1;
9, 4, 2;
10, 4, 2, 1;
11, 5, 2, 1;
12, 5, 3, 1;
13, 6, 3, 1;
14, 6, 3, 2;
15, 7, 4, 2, 1;
16, 7, 4, 2, 1;
17, 8, 4, 2, 1;
18, 8, 5, 3, 1;
19, 9, 5, 3, 1;
20, 9, 5, 3, 2;
21, 10, 6, 3, 2, 1;
22, 10, 6, 4, 2, 1;
23, 11, 6, 4, 2, 1;
24, 11, 7, 4, 2, 1;
25, 12, 7, 4, 3, 1;
26, 12, 7, 5, 3, 1;
27, 13, 8, 5, 3, 2;
28, 13, 8, 5, 3, 2, 1;
...
For n = 10 the 10th row of triangle is 10, 4, 2, 1, so we have that 10^2 - 4^2 + 2^2 - 1^2 = 100 - 16 + 4 - 1 = 87, the same as A024916(10) = 87, the sum of all divisors of all positive integers <= 10.
Illustration of initial terms in the third quadrant:
. y
Row _|
1 _|1|
2 _|2 _|
3 _|3 |1|
4 _|4 _|1|
5 _|5 |2 _|
6 _|6 _|2|1|
7 _|7 |3 |1|
8 _|8 _|3 _|1|
9 _|9 |4 |2 _|
10 _|10 _|4 |2|1|
11 _|11 |5 _|2|1|
12 _|12 _|5 |3 |1|
13 _|13 |6 |3 _|1|
14 _|14 _|6 _|3|2 _|
15 _|15 |7 |4 |2|1|
16 _|16 _|7 |4 |2|1|
17 _|17 |8 _|4 _|2|1|
18 _|18 _|8 |5 |3 |1|
19 _|19 |9 |5 |3 _|1|
20 _|20 _|9 _|5 |3|2 _|
21 _|21 |10 |6 _|3|2|1|
22 _|22 _|10 |6 |4 |2|1|
23 _|23 |11 _|6 |4 |2|1|
24 _|24 _|11 |7 |4 _|2|1|
25 _|25 |12 |7 _|4|3 |1|
26 _|26 _|12 _|7 |5 |3 _|1|
27 _|27 |13 |8 |5 |3|2 _|
28 |28 |13 |8 |5 |3|2|1|
...
T(n,k) is also the number of cells between the k-th vertical line segment (from left to right) and the y-axis in the n-th row of the structure.
Note that the number of horizontal line segments in the n-th row of the structure equals A001227(n), the number of odd divisors of n.
Also the diagram represents the left part of the front view of the pyramid described in A245092. (End)
For n = 12 the symmetric representation of sigma(12) in the fourth quadrant is as shown below: _
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_ _ _ _ _ _| |3 1
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12 5
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For n = 12 and k = 1 the total length of all line segments between the first vertex and the central vertex of the largest Dyck path is equal to 12, so T(12,1) = 12.
For n = 12 and k = 2 the total length of all line segments between the second vertex and the central vertex of the largest Dyck path is equal to 5, so T(12,2) = 5.
For n = 12 and k = 3 the total length of all line segments between the third vertex and the central vertex of the largest Dyck path is equal to 3, so T(12,3) = 3.
For n = 12 and k = 4 the total length of all line segments between the fourth vertex and the central vertex of the largest Dyck path is equal to 1, so T(12,4) = 1.
Hence the 12th row of triangle is [12, 5, 3, 1]. (End)
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MATHEMATICA
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row[n_] := Floor[(Sqrt[8*n + 1] - 1)/2]; f[n_, k_] := Ceiling[(n + 1)/k - (k + 1)/2]; Table[f[n, k], {n, 1, 150}, {k, 1, row[n]}] // Flatten (* Hartmut F. W. Hoft, Apr 07 2014 *)
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PROG
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(PARI) row(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i); \\ Michel Marcus, Mar 27 2014
(Python)
from sympy import sqrt
import math
def T(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2))
for n in range(1, 21): print([T(n, k) for k in range(1, int(math.floor((sqrt(8*n + 1) - 1)/2)) + 1)]) # Indranil Ghosh, Apr 25 2017
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CROSSREFS
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Cf. A000203, A000217, A001227, A196020, A211343, A228813, A231345, A231347, A235794, A236106, A236112, A237270, A237271, A237593, A239660, A245092, A261699, A262626, A286000, A286001, A280850, A280851, A296508, A335616.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A302242
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Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).
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+30
202
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0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 2, 2, 1, 0, 2, 1, 3, 2, 3, 3, 3, 1, 1, 3, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 2, 2, 4, 2, 1, 2, 3, 1, 4, 0, 3, 2, 1, 1, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 1, 4, 1, 1, 3, 2, 2, 3, 1, 4
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OFFSET
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1,7
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COMMENTS
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A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.
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LINKS
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EXAMPLE
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Sequence of finite multisets of finite multisets of positive integers begins: (), (()), ((1)), (()()), ((2)), (()(1)), ((11)), (()()()), ((1)(1)), (()(2)), ((3)), (()()(1)), ((12)), (()(11)), ((1)(2)), (()()()()), ((4)), (()(1)(1)), ((111)), (()()(2)).
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MAPLE
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with(numtheory):
a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
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MATHEMATICA
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primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[PrimeOmega/@primeMS[n]], {n, 100}]
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PROG
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CROSSREFS
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Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A049076
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Number of steps in the prime index chain for the n-th prime.
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+30
64
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1, 2, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1
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OFFSET
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1,2
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COMMENTS
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Let p(k) = k-th prime, let S(p) = S(p(k)) = k, the subscript of p; a(n) = order of primeness of p(n) = 1+m where m is largest number such that S(S(..S(p(n))...)) with m S's is a prime.
The record holders correspond to A007097.
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LINKS
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FORMULA
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Let b(n) = 0 if n is nonprime, otherwise b(n) = k where n is the k-th prime. Then a(n) is the number of times you can apply b to the n-th prime before you hit a nonprime.
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EXAMPLE
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11 is 5th prime, so S(11)=5, 5 is 3rd prime, so S(S(11))=3, 3 is 2nd prime, so S(S(S(11)))=2, 2 is first prime, so S(S(S(S(11))))=1, not a prime. Thus a(5)=4.
Alternatively, a(5) = 4: the 5th prime is 11 and its prime index chain is 11->5->3->2->1->0. a(6) = 1: the 6th prime is 13 and its prime index chain is 13->6->0.
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MAPLE
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if not isprime(n) then
1 ;
else
1+procname(numtheory[pi](n)) ;
end if;
end proc:
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MATHEMATICA
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Table[Length[NestWhileList[PrimePi[#]&, Prime[n], PrimeQ[#]&]]-1, {n, 110}] (* Harvey P. Dale, May 07 2018 *)
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PROG
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(PARI) apply(p->my(s=1); while(isprime(p=primepi(p)), s++); s, primes(100)) \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
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CROSSREFS
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KEYWORD
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nice,nonn,easy
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AUTHOR
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EXTENSIONS
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Additional comments from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2003
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STATUS
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approved
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A006842
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Triangle read by rows: row n gives numerators of Farey series of order n.
(Formerly M0041)
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+30
51
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0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 3, 4, 5, 6, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 5, 6, 7, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 4, 1, 5, 4, 3, 5, 2, 5
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OFFSET
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1,9
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923. See Vol. 1.
Guthery, Scott B. A motif of mathematics. Docent Press, 2011.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
A. O. Matveev, Farey Sequences, De Gruyter, 2017.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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0/1, 1/1;
0/1, 1/2, 1/1;
0/1, 1/3, 1/2, 2/3, 1/1;
0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1;
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1;
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MAPLE
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Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end: seq(numer(Farey(i)), i=1..5); # Peter Luschny, Apr 28 2009
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MATHEMATICA
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Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Flatten[ Table[ Numerator[ Farey[n]], {n, 0, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
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PROG
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(PARI) row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k); ); ); vf = vecsort(Set(vf)); for (i=1, #vf, print1(numerator(vf[i]), ", ")); } \\ Michel Marcus, Jun 27 2014
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CROSSREFS
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KEYWORD
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nonn,nice,frac,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A286001
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A table of partitions into consecutive parts (see Comments lines for definition).
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+30
50
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1, 2, 3, 1, 4, 2, 5, 2, 6, 3, 1, 7, 3, 2, 8, 4, 3, 9, 4, 2, 10, 5, 3, 1, 11, 5, 4, 2, 12, 6, 3, 3, 13, 6, 4, 4, 14, 7, 5, 2, 15, 7, 4, 3, 1, 16, 8, 5, 4, 2, 17, 8, 6, 5, 3, 18, 9, 5, 3, 4, 19, 9, 6, 4, 5, 20, 10, 7, 5, 2, 21, 10, 6, 6, 3, 1, 22, 11, 7, 4, 4, 2, 23, 11, 8, 5, 5, 3, 24, 12, 7, 6, 6, 4, 25, 12, 8, 7, 3, 5
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OFFSET
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1,2
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COMMENTS
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This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms, where the m-th block starts with m, m>=1, and the first element of column k is in row k*(k+1)/2.
The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, but in increasing order, exclusively in the columns where the blocks begin.
More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).
A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.
A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
For a theorem related to this table see A286000.
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LINKS
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EXAMPLE
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Triangle begins:
1;
2;
3, 1;
4, 2;
5, 2;
6, 3, 1;
7, 3, 2;
8, 4, 3;
9, 4, 2;
10, 5, 3, 1;
11, 5, 4, 2;
12, 6, 3, 3;
13, 6, 4, 4;
14, 7, 5, 2;
15, 7, 4, 3, 1;
16, 8, 5, 4, 2;
17, 8, 6, 5, 3;
18, 9, 5, 3, 4;
19, 9, 6, 4, 5;
20, 10, 7, 5, 2;
21, 10, 6, 6, 3, 1;
22, 11, 7, 4, 4, 2;
23, 11, 8, 5, 5, 3;
24, 12, 7, 6, 6, 4;
25, 12, 8, 7, 3, 5;
26, 13, 9, 5, 4, 6;
27, 13, 8, 6, 5, 2;
28, 14, 9, 7, 6, 3, 1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:
. ------------------------------------------------------------------------
Fig: A B C D E F G
. ------------------------------------------------------------------------
. n: 1 2 3 4 5 6 7
Row ------------------------------------------------------------------------
1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; |
2 | | [2];| 2; | 2; | 2; | 2; | 2; |
3 | | | [3],[1];| 3, 1;| 3, 1; | 3, 1; | 3, 1; |
4 | | | 4 ,[2];| [4], 2;| 4, 2; | 4, 2; | 4, 2; |
5 | | | | | [5],[2]; | 5, 2; | 5, 2; |
6 | | | | | 6, [3], 3;| [6], 3, [1];| 6, 3, 1;|
7 | | | | | | 7, 3, [2];| [7],[3], 2;|
8 | | | | | | 8, 4, [3];| 8, [4], 3;|
. ------------------------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts (but with the parts in increasing order) are [6] and [1, 2, 3]. These partitions have 1 and 3 consecutive parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.
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Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:
. --------------------------------------------------------------------
Fig: H I J K
. --------------------------------------------------------------------
. n: 8 9 10 11
Row --------------------------------------------------------------------
1 | 1; | 1; | 1; | 1; |
1 | 2; | 2; | 2; | 2; |
3 | 3, 1; | 3, 1; | 3, 1; | 3, 1; |
4 | 4, 2; | 4, 2; | 4, 2; | 4, 2; |
5 | 5, 2; | 5, 2; | 5, 2; | 5, 2; |
6 | 6, 3, 3;| 6, 3, 1; | 6, 3, 1; | 6, 3, 1; |
7 | 7, 3, 2;| 7, 3, 2; | 7, 3, 2; | 7, 3, 2; |
8 | [8], 4, 1;| 8, 4, 3; | 8, 4, 3; | 8, 4, 3; |
9 | | [9],[4],[2]; | 9, 4, 2; | 9, 4, 2; |
10 | | 10, [5],[3], 1;| [10], 5, 3, [1];| 10, 5, 3, 1;|
11 | | 11, 5, [4], 2;| 11, 5, 4, [2];| [11],[5], 4, 2;|
12 | | | 12, 6, 3, [3];| 12, [6], 3, 3;|
13 | | | 13, 6, 4, [4];| 13, 6, 4, 4;|
. --------------------------------------------------------------------
Figure J: For n = 10 the partitions of 10 into consecutive parts (but with the parts in increasing order) are [10] and [1, 2, 3, 4]. These partitions have 1 and 4 consecutive parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.
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Illustration of initial terms arranged into the diagram of the triangle A237591:
. _
. _|1|
. _|2 _|
. _|3 |1|
. _|4 _|2|
. _|5 |2 _|
. _|6 _|3|1|
. _|7 |3 |2|
. _|8 _|4 _|3|
. _|9 |4 |2 _|
. _|10 _|5 |3|1|
. _|11 |5 _|4|2|
. _|12 _|6 |3 |3|
. _|13 |6 |4 _|4|
. _|14 _|7 _|5|2 _|
. _|15 |7 |4 |3|1|
. _|16 _|8 |5 |4|2|
. _|17 |8 _|6 _|5|3|
. _|18 _|9 |5 |3 |4|
. _|19 |9 |6 |4 _|5|
. _|20 _|10 _|7 |5|2 _|
. _|21 |10 |6 _|6|3|1|
. _|22 _|11 |7 |4 |4|2|
. _|23 |11 _|8 |5 |5|3|
. _|24 _|12 |7 |6 _|6|4|
. _|25 |12 |8 _|7|3 |5|
. _|26 _|13 _|9 |5 |4 _|6|
. _|27 |13 |8 |6 |5|2 _|
. |28 |14 |9 |7 |6|3|1|
...
The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.
.
The connection (described step by step) between the triangle of A299765 and the above geometric diagram is as follows:
.
[1]; [1];
[2]; [2];
[3], [2, 1]; [3], [2, 1];
[4]; [4];
[5], [3, 2]; [5], [3, 2];
[6], [3, 2, 1]; [6], [3, 2, 1];
[7], [4, 3]; [7], [4, 3];
[8]; [8];
[9], [5, 4], [4, 3, 2]; [9], [5, 4], [4, 3, 2];
.
Figure 1. Figure 2.
.
We start with the irregular Then we write the same triangle
triangle of A299765 in which but ordered in columns where the
row n lists the partitions column k lists the partitions of
of n into consecutive parts. n into k consecutive parts.
.
. _ _
1| |1
_ _
2| |2
_ _ _ _ _
3| 2,1| |3 |1
_ _ |2
4| |4
_ _ _ _ _
5| 3,2| |5 |2
_ _ _ _ _ |3 _
6| 3,2,1| |6 |1
_ _ _ _ _ |2
7| 4,3| |7 |3 |3
_ _ |4
8| |8
_ _ _ _ _ _ _ _ _
9| 5,4| 4,3,2| |9 |4 |2
|5 |3
|4
.
Figure 3. Figure 4.
.
Then we draw to the right of Then we rotate each sub-diagram
each partition a vertical 90 degrees counterclockwise.
toothpick and above each part Every horizontal toothpick represents
we draw a horizontal toothpick. the existence of that partition.
. The number of vertical toothpicks
. equals the number of parts.
.
. _ _
_|1 _|1
_|2 _ _|2 _
_|3 |1 _|3 |1
_|4 _|2 _|4 _|2
_|5 |2 _ _|5 |2 _
_|6 _|3|1 _|6 _|3|1
_|7 |3 |2 _|7 |3 |2
_|8 _|4 _|3 _|8 _|4 _|3
|9 |4 |2 |9 |4 |2
|5 |3
|4
.
Figure 5. Figure 6.
.
Then we join the sub-diagrams Finally we erase the parts that
forming staircases (or zig-zag are beyond a certain level (in
paths) that represent the this case beyond the 9th level)
partitions that have the same to make the diagram more standard.
number of parts.
.
The numbers in the k-th staircase (from left to right) are the elements of the k-th column of the triangular array.
There is an infinite family of this kind of triangles, which are related to polygonal numbers and partitions into consecutive parts that differ by d. For more information see the theorems in A285914 and A303300.
Note that if we take two images of the diagram mirroring each other, with the y-axis in the middle of them, then a new diagram is formed, which is symmetric and represents the sequence A237593 as an isosceles triangle. Then if we fold each level (or row) of that isosceles triangle we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n). (End)
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CROSSREFS
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Tables of the same family where the consecutive parts differ by d are A010766 (d=0), this sequence (d=1), A332266 (d=2), A334945 (d=3), A334618(d=4).
Cf. A000217, A001227, A003056, A109814, A196020, A204217, A211343, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A280850, A280851, A285914, A286013, A288529, A288772, A288773, A288774, A296508, A299765, A303300.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A336811
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Irregular triangle read by rows T(n,k) in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive integers A000027, with n >= 1 and k >= 1.
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+30
50
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1, 2, 3, 1, 4, 2, 1, 5, 3, 2, 1, 1, 6, 4, 3, 2, 2, 1, 1, 7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 10, 8, 7, 6, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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1,2
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COMMENTS
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In other words: row n lists A028310(n-1) blocks where the m-th block consists of A187219(m) copies of n - m + [m=1], with n >= 1 and m >= 1, where [] is the Iverson bracket. [Corrected by Paolo Xausa, Feb 10 2023]
All divisors of all terms in row n are also all parts in the last section of the set of partitions of n.
Thus all divisors of all terms of the first n rows of triangle are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n. - Omar E. Pol, Jun 19 2021
The number of k's in row n is equal to A002865(n-k), 1 <= k <= n.
The number of terms >= k in row n is equal to A000041(n-k), 1 <= k <= n.
The number of k's in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000041(n-k), 1 <= k <= n.
The number of terms >= k in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.
First n rows of triangle (or first A000070(n-1) terms of the sequence) in nonincreasing order give the n-th row of A176206. (End)
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LINKS
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EXAMPLE
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Triangle begins:
1;
2;
3, 1;
4, 2, 1;
5, 3, 2, 1, 1;
6, 4, 3, 2, 2, 1, 1;
7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1;
8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1;
9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
...
For n = 6, by definition the length of row 6 is A000041(6-1) = A000041(5) = 7, so the row 6 of triangle has seven terms. Since every column lists the positive integers A000027 so the row 6 is [6, 4, 3, 2, 2, 1, 1].
Then we have that the divisors of the numbers of the 6th row are:
.
6th row of the triangle ----------> 6 4 3 2 2 1 1
3 2 1 1 1
2 1
1
.
There are seven 1's, four 2's, two 3's, one 4 and one 6.
In total there are 7 + 4 + 2 + 1 + 1 = 15 divisors.
On the other hand the last section of the set of the partitions of 6 can be represented in several ways, five of them as shown below:
._ _ _ _ _ _
|_ _ _ | 6 6 6 6
|_ _ _|_ | 3 3 3 3 3 3 3 3
|_ _ | | 4 2 4 2 4 2 4 2
|_ _|_ _|_ | 2 2 2 2 2 2 2 2 2 2 2 2
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
|_| 1 1 1 1
.
Figure 1. Figure 2. Figure 3. Figure 4. Figure 5.
.
In every figure there are seven 1's, four 2's, two 3's, one 4 and one 6, as shown also the 6th row of A182703.
In total there are 7 + 4 + 2 + 1 + 1 = A138137(6) = 15 parts in every figure.
Figure 5 is an arrangement that shows the correspondence between divisors and parts since the columns give the divisors of the terms of 6th row of triangle.
Finally we can see that all divisors of all numbers in the 6th row of the triangle are the same positive integers as all parts in the last section of the set of the partitions of 6.
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MATHEMATICA
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A336811[row_]:=Flatten[Table[ConstantArray[row-m, PartitionsP[m]-PartitionsP[m-1]], {m, 0, row-1}]];
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PROG
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(PARI) f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--; ); 1+s; }
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n, k), ", "); ); print; ); } \\ Michel Marcus, Jan 13 2021
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CROSSREFS
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Cf. A000007, A000041, A027750, A028310, A002865, A133735, A135010, A138121, A138137, A182703, A187219, A207378, A221529, A336812, A339278, A340035, A340061, A346741.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A194832
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Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {r}, {2r}, ..., {nr}, where r= -tau = -(1+sqrt(5))/2.
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+30
47
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1, 1, 2, 3, 1, 2, 3, 1, 4, 2, 3, 1, 4, 2, 5, 3, 6, 1, 4, 2, 5, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 1, 9, 4, 7, 2, 5, 8, 3, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 13, 8, 3, 11
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OFFSET
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1,3
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COMMENTS
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Every irrational number r generates a triangular array in the manner exemplified here. Taken as a sequence, the numbers comprise a fractal sequence f which induces a second (rectangular) array whose n-th row gives the positions of n in f. Denote these by Array1 and Array2. As proved elsewhere, Array2 is an interspersion. (Every row intersperses every other row except for initial terms.) Taken as a sequence, Array2 is a permutation, Perm1, of the positive integers; let Perm2 denote its inverse permutation.
Examples:
r................Array1....Array2....Perm2
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REFERENCES
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C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997), 157-168.
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LINKS
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EXAMPLE
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Fractional parts: {-r}=-0.61..;{-2r}=-0.23..;{-3r}=-0.85..;{-4r}=-0.47..; thus, row 4 is (3,1,4,2) because {-3r} < {-r} < {-4r} < {-2r}. [corrected by Michel Dekking, Nov 30 2020]
First nine rows:
1
1 2
3 1 2
3 1 4 2
3 1 4 2 5
3 6 1 4 2 5
3 6 1 4 7 2 5
8 3 6 1 4 7 2 5
8 3 6 1 9 4 7 2 5
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MATHEMATICA
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r = -GoldenRatio;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]]
TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]
row[n_] := Position[f, n];
u = TableForm[Table[row[n], {n, 1, 20}]]
g[n_, k_] := Part[row[n], k];
p = Flatten[Table[g[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A194833 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A194834 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A338912
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Lesser prime index of the n-th semiprime.
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+30
43
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1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 5, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 6, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1
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1,3
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COMMENTS
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A semiprime is a product of any two prime numbers. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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FORMULA
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EXAMPLE
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The semiprimes are:
2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the lesser prime factors are:
2, 2, 3, 2, 2, 3, 3, 2, 5, 2, ...
with indices:
1, 1, 2, 1, 1, 2, 2, 1, 3, 1, ...
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MATHEMATICA
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Table[Min[PrimePi/@First/@FactorInteger[n]], {n, Select[Range[100], PrimeOmega[#]==2&]}]
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CROSSREFS
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A084126 is the lesser prime factor (not index).
A128301 lists positions of first appearances of each positive integer.
A001221 counts distinct prime indices.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A270650
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Min(i, j), where p(i)*p(j) is the n-th term of A006881.
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+30
42
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1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1, 1, 4, 2, 3, 1
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OFFSET
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1,4
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LINKS
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EXAMPLE
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A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes. The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (1,1,1,2).
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MATHEMATICA
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mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}] (* A096916 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}] (* A070647 *)
Map[PrimePi[FactorInteger[#][[1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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1, 1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 5, 1, 6, 4, 2, 7, 3, 5, 8, 1, 9, 6, 4, 10, 2, 11, 7, 3, 12, 5, 8, 13, 1, 14, 9, 6, 15, 4, 10, 16, 2, 17, 11, 7, 18, 3, 19, 12, 5, 20, 8, 13, 21, 1, 22, 14, 9, 23, 6, 15, 24, 4, 25, 10, 16, 26, 2, 27, 17, 11, 28, 7, 18, 29, 3, 30, 19, 12, 31, 5, 32, 20, 8, 33
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OFFSET
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1,4
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REFERENCES
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D. R. Morrison, A Stolarsky Array of Wythoff Pairs, A Collection of Manuscripts Related to the Fibonacci Sequence, edited by V. E. Hoggatt Jr., M. Bicknell-Johnson, published by The Fibonacci Association, (1980) pp. 134-136. - Casey Mongoven, Sep 10 2011
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LINKS
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FORMULA
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EXAMPLE
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As a fractal sequence, if each first occurrence of each n is deleted, then the resulting sequence is the same as the original. For the fractal sequence of the Wythoff array, see A003603.
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MAPLE
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A035506 := proc(r, c) local tau, a, b, d, i ; tau := (1+sqrt(5))/2 ; a := floor( r*(1+tau)-tau/2) ; b := round(a*tau) ; if c = 1 then RETURN(a) ; else if c =2 then RETURN(b) ; else for i from 1 to c-2 do d := a+b ; a := b; b := d ; od: RETURN(d) ; fi ; fi ; end:
A133299 := proc(n) local row, col ; for row from 1 do for col from 1 do stola := A035506(row, col) ; if stola = n then RETURN(row) ; elif stola > n then break ; fi ; od: od: end:
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MATHEMATICA
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A035506[r_, c_] := Module[{tau = GoldenRatio, a, b, d, i}, a = Floor[r*(1 + tau) - tau/2]; b = Round[a*tau]; If[c == 1, Return[a], If[c == 2, Return[b], For[i = 1, i <= c - 2, i++, d = a + b; a = b; b = d]; Return[d]]]];
a[n_] := Module[{row, col}, For[row = 1, True, row++, For[col = 1, True, col++, stola = A035506[row, col] ; If[stola == n, Return[row], If[stola > n, Break[]]]]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Definition now conforms to others; comment replaced - Clark Kimberling, Oct 29 2009
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STATUS
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approved
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