A260739 -id:A260739 - cp's OEIS frontend

A254100 Postludic numbers: Second column of Ludic array A255127.

4, 9, 19, 31, 55, 73, 101, 145, 167, 205, 253, 293, 317, 355, 413, 473, 521, 569, 623, 677, 737, 763, 833, 917, 983, 1027, 1051, 1121, 1171, 1273, 1337, 1411, 1471, 1571, 1619, 1663, 1681, 1807, 1957, 1991, 2087, 2113, 2171, 2245, 2275, 2335, 2401, 2497, 2593, 2713, 2771, 2831, 2977, 3047, 3113

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Column 2 of A255127. (Row 2 of A255129). Positions of 2's in A260739.
Subsequence of A192607, A302036 and A302038.
Cf. A276576, A276606 (first differences).
Cf. also A001248, A219178.

rows = 100; cols = 2; t = Range[2, 10^4]; r = {1}; n = 1; While[n <= rows, k = First[t]; AppendTo[r, k]; t0 = t; t = Drop[t, {1, -1, k}]; ro[n++] = Complement[t0, t][[1 ;; cols]]]; A = Array[ro, rows]; Table[A[[n, 2]], {n, 1, rows} ] (* Jean-François Alcover, Mar 14 2016, after Ray Chandler *)

(define (A254100 n) (A255127bi n 2)) ;; A255127bi given in A255127.

a(n) = A255407(A001248(n)).

A260738 Row index to A255127: a(1) = 0; for n > 1, a(n) = number of the stage where n is removed in the sieve which produces Ludic numbers.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 1, 3, 1, 2, 1, 8, 1, 9, 1, 2, 1, 10, 1, 4, 1, 2, 1, 3, 1, 11, 1, 2, 1, 12, 1, 13, 1, 2, 1, 14, 1, 3, 1, 2, 1, 15, 1, 5, 1, 2, 1, 4, 1, 16, 1, 2, 1, 3, 1, 17, 1, 2, 1, 18, 1, 6, 1, 2, 1, 19, 1, 3, 1, 2, 1, 20, 1, 4, 1, 2, 1, 21, 1, 22, 1, 2, 1, 3, 1, 23, 1, 2, 1, 7, 1, 5, 1, 2, 1, 24, 1, 3, 1, 2, 1, 4, 1, 25, 1, 2, 1, 26, 1

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10105

Row index to array A255127.
Cf. A003309, A254100.
Cf. A260739 (corresponding column index).
Cf. A055396, A260438 for row indices to other arrays similar to A255127.
Differs from A055396 for the first time at n=19.

(define (A260738 n) (cond ((= 1 n) 0) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255127bi row col) n) (if (= (A255127bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255127bi given in A255127.

Other identities. For all n >= 1:
a(A003309(n)) = n-1. [In Ludic sieve A003309(k+1) (i.e., the k-th Ludic number after 1) is the first among the numbers removed at stage k.]
a(2n) = 1. [All even numbers are removed at the stage one of the sieve.]
a(A016945(n)) = 2, a(A255413(n)) = 3, a(A255414(n)) = 4, ..., a(A255419(n)) = 9.
a(A254100(n)) = n.
For all n >= 2:
A255127(a(n), A260739(n)) = n.

A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 21, 22, 19, 24, 23, 26, 27, 28, 29, 30, 49, 32, 33, 34, 35, 36, 31, 50, 39, 40, 37, 42, 41, 44, 45, 38, 43, 48, 55, 46, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 98, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 62, 75, 100, 67, 78, 85, 80, 81

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2244
Index entries for sequences that are permutations of the natural numbers

Inverse: A269171.
Cf. A000035, A003309, A008578, A083221, A250469, A260738, A260739, A269380.
Related or similar permutations: A260741, A260742, A269356, A269358, A255422.
Cf. also A269394 (a(3n)/3) and A269396.
Differs from A255408 for the first time at n=38, where a(38) = 50, while A255408(38) = 38.

a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, A250469(a(A269380(n))).
a(1) = 1, for n > 1, a(n) = A083221(A260738(n), a(A260739(n))).
As a composition of other permutations:
a(n) = A252755(A269386(n)).
a(n) = A252753(A269388(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]

A260439 Column index to A255551: a(1) = 0; for n > 1: if n is Lucky number then a(n) = 1, otherwise for a(2k) = k, and for odd unlucky numbers, a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 4, 1, 5, 3, 6, 1, 7, 1, 8, 4, 9, 2, 10, 1, 11, 5, 12, 1, 13, 2, 14, 6, 15, 1, 16, 1, 17, 7, 18, 1, 19, 3, 20, 8, 21, 1, 22, 2, 23, 9, 24, 1, 25, 1, 26, 10, 27, 2, 28, 3, 29, 11, 30, 4, 31, 1, 32, 12, 33, 1, 34, 1, 35, 13, 36, 1, 37, 1, 38, 14, 39, 1, 40, 5, 41, 15, 42, 2, 43, 1, 44, 16, 45, 4, 46, 1, 47, 17, 48, 3, 49, 1, 50, 18, 51, 6, 52, 1

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

a(1) = 0, because 1 is outside of A255551 array proper.

Antti Karttunen, Table of n, a(n) for n = 1..10003

Cf. A000959, A050505, A145649, A255543, A255551.
Cf. also A260438 (corresponding row index).
Cf. A078898, A246277, A260429, A260437, A260739 for column indices to other arrays similar to A255551.

(define (A260439 n) (cond ((= 1 n) 0) ((not (zero? (A145649 n))) 1) ((even? n) (/ n 2)) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) (+ 1 col) (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(2n) = n.
Also, for all n >= 2:
A255551(A260438(n), a(n)) = n.
a(A219178(n)) = 2.

A302034 A028234 analog for a factorization process based on the Ludic sieve (A255127); Discard all instances of the (smallest) Ludic factor A272565(n) from n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 7, 7, 1, 15, 1, 1, 5, 17, 7, 9, 1, 19, 11, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 19, 13, 1, 27, 1, 7, 7, 29, 11, 15, 1, 31, 13, 1, 11, 33, 1, 17, 5, 35, 1, 9, 1, 37, 17, 19, 1, 39, 7, 5, 11, 41, 1, 21, 1, 43, 35, 11, 1, 45, 1, 23, 1, 47, 13, 3, 1, 49, 23, 25, 1, 51, 13, 13, 19

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the Ludic factor (A272565) of each term gives a sequence of distinct Ludic numbers (A003309) in ascending order, while applying A302035 to the same terms gives the corresponding "exponents" of these Ludic factors in this nonstandard "Ludic factorization of n", unique for each natural number n >= 1. Permutation pair A302025/A302026 maps between this Ludic factorization and the ordinary prime factorization of n. See also comments and examples in A302032.

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A003309, A255127, A260738, A260739, A269379, A269380, A302025, A302026, A302032, A302035.
Cf. A302036 (gives the positions of 1's).
Cf. also A028234, A302044.

\\ Assuming A269379 and its inverse A269380 have been precomputed, then the following is reasonably fast:
A302034(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A269379(n); k--); (n));

For n > 1, a(n) = A269379^(r)(A000265(A260739(n))), where r = A260738(n)-1 and A269379^(r)(n) stands for applying r times the map x -> A269379(x), starting from x = n.

a(n) = A302025(A028234(A302026(n))).

A260429 Column index to A255545: if n is Lucky number, then a(n) = 1, otherwise a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 5, 1, 6, 3, 7, 1, 8, 1, 9, 4, 10, 2, 11, 1, 12, 5, 13, 1, 14, 2, 15, 6, 16, 1, 17, 1, 18, 7, 19, 1, 20, 3, 21, 8, 22, 1, 23, 2, 24, 9, 25, 1, 26, 1, 27, 10, 28, 2, 29, 3, 30, 11, 31, 4, 32, 1, 33, 12, 34, 1, 35, 1, 36, 13, 37, 1, 38, 1, 39, 14, 40, 1, 41, 5, 42, 15, 43, 2, 44, 1, 45, 16, 46, 4, 47, 1, 48, 17, 49, 3, 50, 1, 51, 18, 52, 6, 53, 1

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10003

One more than A260437.
Cf. A000959, A050505, A145649, A219178, A255543, A255545.
Cf. also A260438 (corresponding row index).
Cf. A078898, A246277, A260439, A260739 for column indices to other arrays similar to A255545.

(define (A260429 n) (cond ((not (zero? (A145649 n))) 1) ((even? n) (+ 1 (/ n 2))) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) (+ 1 col) (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(n) = 1 + A260437(n).
Iff A145649(n) = 1, then a(n) = 1.
a(2n) = n+1. [Even numbers are removed at the stage one of the sieve, after 1 which is also removed in the beginning.]
a(A219178(n)) = 2.
A255545(A260438(n), a(n)) = n.

cp's OEIS Frontend

A254100 Postludic numbers: Second column of Ludic array A255127.

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A260738 Row index to A255127: a(1) = 0; for n > 1, a(n) = number of the stage where n is removed in the sieve which produces Ludic numbers.

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A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

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A260439 Column index to A255551: a(1) = 0; for n > 1: if n is Lucky number then a(n) = 1, otherwise for a(2k) = k, and for odd unlucky numbers, a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

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Formula

A302034 A028234 analog for a factorization process based on the Ludic sieve (A255127); Discard all instances of the (smallest) Ludic factor A272565(n) from n.

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A260429 Column index to A255545: if n is Lucky number, then a(n) = 1, otherwise a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

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