editing

approved

1) take the 1-st 1st element from natural numbers (A000027): 1; remaining set is 2,3,4,5,6,7,8,9,10,...; S1={1}

2) take the 1-st 1st element from the remaining set: 2; remaining set is 3,4,5,6,7,8,9,10,...; take the 2-nd 2nd element from the remaining set: 4; remaining set is 3,5,6,7,8,9,10,...; S2={2,4}

3) take the 1-st 1st element from the remaining set: 3; remaining set is 5,6,7,8,9,10,...; take the 3-rd 3rd element from the remaining set: 7; remaining set is 5,6,8,9,10,11,12,...; take the 7-th 7th element from the remaining set: 12; remaining set is 5,6,8,9,10,11,13,14,15,16,17,18,19,20,..; S3={3,7,12}

4) take the 1-st 1st element from the remaining set: 5; remaining set is 6,8,9,10,11,13,14,15,16,17,18,19,20,..; take the 5-th 5th element from the remaining set: 11; remaining set is 6,8,9,10,13,14,15,16,17,18,19,20,..; take the 11-th 11th element from the remaining set: 19; remaining set is 6,8,9,10,13,14,15,16,17,18,20,..; take the 19-th 19th element from the remaining set: 28; remaining set is 6,8,9,10,13,14,15,16,17,18,20,21,22,23,24,25,26,27,29,30,31,...;

approved

editing

_Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), _, May 14 2008

Edited, extended, keyword tabl and Scheme-code added by _Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), _, Oct 05 2009

<a href="/Sindx_index/Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

A. Karttunen, <a href="/A138612/b138612.txt">Table of n, a(n) for n = 1..5050</a>

<a href="/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

nonn,tabl,new

Concatenation Permutation of subsequencesnatural numbers generated with the sieve algorithm described in the comment lines.

1, 2, 4, 3, 7, 12, 5, 11, 19, 28, 6, 15, 26, 39, 53, 8, 20, 35, 52, 71, 91, 9, 23, 42, 64, 88, 114, 141, 10, 27, 49, 76, 106, 138, 172, 207, 13, 33, 60, 93, 129, 168, 210, 253, 297, 14, 37, 68, 105, 148, 194, 243, 294, 347, 401, 16, 43, 79, 122, 171, 225, 282, 342

Sieve proceeds as:

1) take the 1-st element from natural numbers (A000027): 1; remaining set is 2,3,4,5,6,7,8,9,10,...; S1={1}

2) take the 1-st element from the remaining set: 2; remaining set is 3,4,5,6,7,8,9,10,...; take the 2-nd element from the remaining set: 4; remaining set is 3,5,6,7,8,9,10,...; S2={2,4}

3) take the 1-st element from the remaining set: 3; remaining set is 5,6,7,8,9,10,...; take the 3-rd element from the remaining set: 7; remaining set is 5,6,8,9,10,11,12,...; take the 7-th element from the remaining set: 12; remaining set is 5,6,8,9,10,11,13,14,15,16,17,18,19,20,..; S3={3,7,12}

4) take the 1-st element from the remaining set: 5; remaining set is 6,8,9,10,11,13,14,15,16,17,18,19,20,..; take the 5-th element from the remaining set: 11; remaining set is 6,8,9,10,13,14,15,16,17,18,19,20,..; take the 11-th element from the remaining set: 19; remaining set is 6,8,9,10,13,14,15,16,17,18,20,..; take the 19-th element from the remaining set: 28; remaining set is 6,8,9,10,13,14,15,16,17,18,20,21,22,23,24,25,26,27,29,30,31,...;

thus S4={5,11,19,28}.

The sequence is concatenation of such subsequences S1,S2,S3,S4,S5,...,Sn, ..., where each subsequence consists of n nondecreasing terms. Alternatively, these can be viewed as rows of a triangular table.

A. Karttunen, <a href="b138612.txt">Table of n, a(n) for n = 1..5050</a>

<a href="Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

1. The sequence is formed by concatenating subsequences S1,S2,..., each of finite length. 2. The subsequence S1 consists of the element 1. 3. The subsequence Sn consists of n elements. 4.Each subsequence is nondecreasing 5. The difference between two consecutive elements in the same subsequence is varying, but >=1.

given set of natural numbers : 1,2,3,4,5,6,7,8,9,10,...

there are n

1)take the 1-st element from natural numbers : 1; remaining set is 2,3,4,5,6,7,8,9,10,...; S1={1}

2)take the 1-st element from the remaining set : 2; remaining set is 3,4,5,6,7,8,9,10,... ; take the 2-nd element from the remaining set : 4 ; remaining set is 3,5,6,7,8,9,10,...; S2={2,4}

3)take the 1-st element from the remaining set : 3; remaining set is 5,6,7,8,9,10,...; take the 3-rd element from the remaining set : 7 ; remaining set is 5,6,8,9,10,11,12,... ; take the 7-th element from the remaining set : 12; remaining set is 5,6,8,9,10,11,13,14,15,16,17,18,19,20,..; S3={3,7,12}

4)take the 1-st element from the remaining set: 5; remaining set is 6,8,9,10,11,13,14,15,16,17,18,19,20,..; take the 5-th element from the remaining set : 11 ; remaining set is 6,8,9,10,13,14,15,16,17,18,19,20,..; take the 11-th element from the remaining set : 19; remaining set is 6,8,9,10,13,14,15,16,17,18,20,..; take the 19-th element from the remaining set : 28; remaining set is 6,8,9,10,13,14,15,16,17,18,20,21,22,23,24,25,26,27,29,30,31,...;

S4={5,11,19,28}

...

The sequence is concatenation of subsequences S1,S2,S3,S4,...

which gives : 1,2,4,3,7,12,5,11,19,28,6,15,26,...

(MIT Scheme:)

(define (A138612 n) (if (< n 3) n (let loop ((k (if (zero? (A002262 (-1+ n))) 1 (A138612 (-1+ n)))) (i 1)) (cond ((not-lte? (A166017 i) (-1+ n)) (if (= 1 k) i (loop (-1+ k) (1+ i)))) (else (loop k (1+ i)))))))

(define (not-lte? a b) (cond ((not (number? a)) #t) (else (> a b))))

Inverse: A166017. Left edge A166018, Right edge: A166019, Row sums: A166020. Cf. A001614A138606-A138609.

easy,nonn,uned,new

nonn,tabl

Edited, extended, keyword tabl and Scheme-code added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 05 2009

1. The sequence is formed by concatenating subsequences S1,S2,..., each of finite length. 2. The subsequence S1 consists of the element 1. 3. The subsequence Sn consists of n elements. 4.Each subsequence is nondecreasing 5. The difference between two consecutive elements in the same subsequence is varying, but >=1.

Cf. A001614.

easy,nonn,uned,new

Concatenation of subsequences.

1, 2, 4, 3, 7, 12, 5, 11, 19, 28, 6, 15, 26

1,2

1.The sequence is formed by concatenating subsequences S1,S2,..., each of finite length. 2.The subsequence S1 consists of the element 1. 3.The subsequence Sn consists of n elements. 4.Each subsequence is nondecreasing 5.The difference between two consecutive elements in the same subsequence is varying, but >=1.

given set of natural numbers : 1,2,3,4,5,6,7,8,9,10,...

there are n

1)take the 1-st element from natural numbers : 1; remaining set is 2,3,4,5,6,7,8,9,10,...; S1={1}

2)take the 1-st element from the remaining set : 2; remaining set is 3,4,5,6,7,8,9,10,... ; take the 2-nd element from the remaining set : 4 ; remaining set is 3,5,6,7,8,9,10,...; S2={2,4}

3)take the 1-st element from the remaining set : 3; remaining set is 5,6,7,8,9,10,...; take the 3-rd element from the remaining set : 7 ; remaining set is 5,6,8,9,10,11,12,... ; take the 7-th element from the remaining set : 12; remaining set is 5,6,8,9,10,11,13,14,15,16,17,18,19,20,..; S3={3,7,12}

4)take the 1-st element from the remaining set: 5; remaining set is 6,8,9,10,11,13,14,15,16,17,18,19,20,..; take the 5-th element from the remaining set : 11 ; remaining set is 6,8,9,10,13,14,15,16,17,18,19,20,..; take the 11-th element from the remaining set : 19; remaining set is 6,8,9,10,13,14,15,16,17,18,20,..; take the 19-th element from the remaining set : 28; remaining set is 6,8,9,10,13,14,15,16,17,18,20,21,22,23,24,25,26,27,29,30,31,...;

S4={5,11,19,28}

...

The sequence is concatenation of subsequences S1,S2,S3,S4,...

which gives : 1,2,4,3,7,12,5,11,19,28,6,15,26,...

Cf.A001614.

easy,nonn,uned

Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), May 14 2008

approved