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Search: a252460 -id:a252460
Displaying 1-5 of 5 results found. page 1
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A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)). +20
17
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 21, 22, 25, 24, 19, 26, 27, 28, 29, 30, 37, 32, 33, 34, 35, 36, 41, 38, 39, 40, 43, 42, 47, 44, 45, 46, 53, 48, 31, 50, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 71, 62, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 89, 74, 75, 76, 59, 78, 91, 80, 81 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) tells which number in Ludic array A255127 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.
Equally: a(n) tells which number in array A255129 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(n) = A255127(A252460(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes even numbers.]
a(3n) = 3n. [Fixes multiples of three.]
a(A008578(n)) = A003309(n). [Maps noncomposites to Ludic numbers.]
a(A001248(n)) = A254100(n). [Maps squares of primes to "postludic numbers".]
a(A084967(n)) = a(5*A007310(n)) = A007310((5*n)-3) = A255413(n). [Maps A084967 to A255413.]
(And similarly between other columns and rows of A083221 and A255127.)
EXAMPLE
A083221(8,1) = 19 and A255127(8,1) = 23, thus a(19) = 23.
A083221(9,1) = 23 and A255127(9,1) = 25, thus a(23) = 25.
A083221(3,2) = 25 and A255127(3,2) = 19, thus a(25) = 19.
PROG
(define (A255407 n) (A255127 (A252460 n)))
CROSSREFS
Cf. A001248, A003309, A007310, A008578, A083221, A084967, A252460, A254100, A255413, A255127.
Inverse: A255408.
Similar permutations: A249818.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 22 2015
STATUS
approved
A255553 Permutation of natural numbers: a(n) = A255551(A252460(n)). +20
9
1, 2, 3, 4, 7, 6, 9, 8, 5, 10, 13, 12, 15, 14, 11, 16, 21, 18, 25, 20, 17, 22, 31, 24, 19, 26, 23, 28, 33, 30, 37, 32, 29, 34, 39, 36, 43, 38, 35, 40, 49, 42, 51, 44, 41, 46, 63, 48, 27, 50, 47, 52, 67, 54, 61, 56, 53, 58, 69, 60, 73, 62, 59, 64, 81, 66, 75, 68, 65, 70, 79, 72, 87, 74, 71, 76, 57, 78, 93, 80, 77, 82, 99, 84, 103, 86, 83, 88, 105, 90 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) tells which number in array A255551, constructed from Lucky sieve, is at the same position where n is in array A083221, constructed from the sieve of Eratosthenes. As both arrays have A005843 (even numbers) as their topmost row, this permutation fixes all of them.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..6840
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(n) = A255551(A252460(n)).
Other identities:
a(2n) = 2n. [Fixes even numbers.]
For all n >= 1, a(A083141(n)) = A255550(n).
For all n >= 2, a(A000040(n)) = A000959(n).
For all n >= 2, a(A001248(n)) = A219178(n).
PROG
(Scheme) (define (A255553 n) (A255551 (A252460 n)))
CROSSREFS
Inverse: A255554.
Similar or related permutations: A255407, A255408, A249817, A249818, A252460, A255551.
Cf. A000040, A000959, A001248, A083141, A219178, A255550.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 26 2015
STATUS
approved
A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... +10
85
2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015
LINKS
Antti Karttunen, Table of n, a(n) for n = 2..3487; the first 83 antidiagonals of the array, flattened
Index entries for sequences that are permutations of the natural numbers
EXAMPLE
The top left corner of the array:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26
3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75
5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185
7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329
11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583
13, 169, 221, 247, 299, 377, 403, 481, 533, 559, 611, 689, 767
17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037
19, 361, 437, 551, 589, 703, 779, 817, 893, 1007, 1121, 1159, 1273
23, 529, 667, 713, 851, 943, 989, 1081, 1219, 1357, 1403, 1541, 1633
29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
...
MATHEMATICA
lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)
PROG
(Scheme, with Antti Karttunen's IntSeq-library)
(define (A083221 n) (if (<= n 1) n (A083221bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Gives 1 for 1 and then the terms of this square array: (A083221 2) = 2, (A083221 3) = 4, etc.
(define (A083221bi row col) ((rowfun_n_for_A083221 row) col))
(definec (rowfun_n_for_A083221 n) (if (= 1 n) (lambda (n) (+ n n)) (let ((rowfun_of_Esieve (rowfun_n_for_Esieve n)) (prime (A000040 n))) (COMPOSE rowfun_of_Esieve (MATCHING-POS 1 1 (lambda (i) (zero? (modulo (rowfun_of_Esieve i) prime))))))))
(definec (A000040 n) ((rowfun_n_for_Esieve n) 1))
(definec (rowfun_n_for_Esieve n) (if (= 1 n) (lambda (n) (+ 1 n)) (let* ((prevrowfun (rowfun_n_for_Esieve (- n 1))) (prevprime (prevrowfun 1))) (COMPOSE prevrowfun (NONZERO-POS 1 1 (lambda (i) (modulo (prevrowfun i) prevprime)))))))
;; Antti Karttunen, Dec 19 2014
CROSSREFS
Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Column 1: A000040, Column 2: A001248.
Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.
KEYWORD
nonn,tabl,look
AUTHOR
Yasutoshi Kohmoto, Jun 05 2003
EXTENSIONS
More terms from Hugo Pfoertner, Jun 13 2003
STATUS
approved
A252752 Inverse permutation to sequence A246278 when it is considered as a permutation of natural numbers (with assumption that a(1) = 1). +10
5
1, 2, 4, 3, 7, 5, 11, 8, 6, 12, 16, 17, 22, 23, 9, 30, 29, 38, 37, 47, 18, 57, 46, 68, 10, 80, 13, 93, 56, 107, 67, 122, 31, 138, 14, 155, 79, 173, 69, 192, 92, 212, 106, 233, 24, 255, 121, 278, 15, 302, 94, 327, 137, 353, 25, 380, 156, 408, 154, 437, 172, 467, 58, 498, 40, 530, 191, 563, 193, 597, 211, 632, 232, 668, 48, 705, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(1) = 1; for n>1: a(n) = 1 + A246276(n-1).
As a composition of related permutations:
a(n) = A253562(A122111(n)).
a(n) = 1 + A253552(A156552(n)).
PROG
(Scheme, two versions)
(define (A252752 n) (if (<= n 1) n (let ((x (A055396 n)) (y (A246277 n))) (+ 1 (* (/ 1 2) (- (expt (+ x y) 2) x y y y -2))))))
(define (A252752 n) (if (<= n 1) n (+ 1 (A246276 (- n 1)))))
CROSSREFS
Inverse of array A246278 considered as a permutation of natural numbers with prepended a(1) = 1.
Cf. A055396, A246277.
Related permutations A122111, A156552, A246276, A253552, A253562.
Differs from A252460 for the first time at n=21, where a(21) = 18, while A252460(21) = 13.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 03 2015
STATUS
approved
A255128 Inverse permutation to A255127. +10
4
1, 2, 4, 3, 7, 5, 11, 8, 6, 12, 16, 17, 22, 23, 9, 30, 29, 38, 10, 47, 13, 57, 37, 68, 46, 80, 18, 93, 56, 107, 15, 122, 24, 138, 14, 155, 67, 173, 31, 192, 79, 212, 92, 233, 39, 255, 106, 278, 19, 302, 48, 327, 121, 353, 21, 380, 58, 408, 20, 437, 137, 467, 69, 498, 25, 530, 154, 563, 81, 597, 172, 632, 28, 668, 94, 705, 191, 743, 32 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..79.
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(n) = A252460(A255408(n)).
CROSSREFS
Inverse: A255127 (with assumed silent a(1)=1).
Cf. A252460, A255408, A255130.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 22 2015
STATUS
approved
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Last modified August 29 09:09 EDT 2024. Contains 375511 sequences. (Running on oeis4.)