A245607 -id:A245607 - cp's OEIS frontend

A245705 Permutation of natural numbers: a(n) = A245607(A245708(n)).

1, 2, 3, 4, 5, 6, 9, 8, 13, 10, 7, 12, 37, 18, 19, 16, 69, 26, 17, 20, 33, 14, 277, 24, 213, 74, 11, 36, 73, 38, 71, 32, 45, 138, 25, 52, 1109, 34, 457, 40, 173, 66, 43, 28, 549, 554, 3351, 48, 77, 426, 21, 148, 53, 22, 135, 72, 125, 146, 651, 76, 273, 142, 55, 64

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

The even bisection halved gives the sequence back. The odd bisection incremented by one and halved gives A245711.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245706.
Fixed points: A245709.
Cf. A245607, A245708, A245605, A245711.

A245705(n) = A245607(A245708(n));
for(n=1, 10001, write("b245705.txt", n, " ", A245705(n)));
\\ Other code as in A245607 and A245708.

(define (A245705 n) (A245607 (A245708 n)))

a(n) = A245607(A245708(n)).

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 21, 5, 34, 23, 53, 6, 59, 29, 20, 1, 33, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16

Offset: 1

Vladeta Jovovic, Oct 30 2001

From Antti Karttunen, May 12 2014: (Start)
a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]
Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.
(End)
From Antti Karttunen, Dec 18-21 2014: (Start)
When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).
For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.
(End)
Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

			a(20) = a(2^2*5) = a(2^2)*a(5) = prevprime(5) = 3.

Harry J. Smith (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064216 (odd bisection), A003961 (inverse), A151799.
Cf. A000040, A008578, A027746, A032742, A049084, A052126, A055396, A061395, A112798, A249817, A249818.
Other sequences whose definition involve or are some other way related with this sequence: A105560, A108951, A118306, A122111, A156552, A163511, A200746, A241909, A243070, A243071, A243072, A243073, A244319, A245605, A245607, A246165, A246266, A246268, A246277, A246278, A246361, A246362, A246371, A246372, A246373, A246374, A246376, A246380, A246675, A246682, A249745, A250470.
Similar prime-shifts towards smaller numbers: A252461, A252462, A252463.

a064989 1 = 1
a064989 n = product $ map (a008578 . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A064989 n) (if (= 1 n) n (apply * (map (lambda (k) (if (zero? k) 1 (A000040 k))) (map -1+ (map A049084 (factor n)))))))
;; Antti Karttunen, May 12 2014
(definec (A064989 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A064989 (A032742 n))))) ;; One based on given recurrence and utilizing memoizing definec-macro.
(definec (A064989 n) (cond ((= 1 n) n) ((even? n) (A064989 (/ n 2))) (else (A163511 (/ (- (A243071 n) 1) 2))))) ;; Corresponds to one of the alternative formulas, but is very unpractical way to compute this sequence. - Antti Karttunen, Dec 18 2014

q:= proc(p) prevprime(p) end proc: q(2):= 1:
[seq(mul(q(f[1])^f[2], f = ifactors(n)[2]), n = 1 .. 1000)]; # Robert Israel, Dec 21 2014

Table[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, Jan 04 2016 *)

{ for (n=1, 1000, f=factor(n)~; a=1; j=1; if (n>1 && f[1, 1]==2, j=2); for (i=j, length(f), a*=precprime(f[1, i] - 1)^f[2, i]); write("b064989.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((p=f[i,1]) % 2, f[i,1] = precprime(p-1), f[i,1] = 1);); factorback(f);} \\ Michel Marcus, Dec 18 2014

A064989(n)=factorback(Mat(apply(t->[max(precprime(t[1]-1),1),t[2]],Vec(factor(n)~))~)) \\ M. F. Hasler, Dec 29 2014

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

from math import prod
from sympy import prevprime, factorint
def A064989(n): return prod(prevprime(p)**e for p, e in  factorint(n>>(~n&n-1).bit_length()).items()) # Chai Wah Wu, Jan 05 2023

From Antti Karttunen, Dec 18 2014: (Start)
If n = product A000040(k)^e(k) then a(n) = product A008578(k)^e(k) [where A000040(n) gives the n-th prime, and A008578(n) gives 1 for 1 and otherwise the (n-1)-th prime].
a(1) = 1; for n > 1, a(n) = A008578(A055396(n)) * a(A032742(n)). [Above formula represented as a recurrence. Cf. A252461.]
a(1) = 1; for n > 1, a(n) = A008578(A061395(n)) * a(A052126(n)). [Compare to the formula of A252462.]
This prime-shift operation is used in the definitions of many other sequences, thus it can be expressed in many alternative ways:
a(n) = A200746(n) / n.
a(n) = A242424(n) / A105560(n).
a(n) = A122111(A122111(n)/A105560(n)) = A122111(A052126(A122111(n))). [In A112798-partition context: conjugate, remove the largest part (the largest prime factor), and conjugate again.]
a(1) = 1; for n > 1, a(2n) = a(n), a(2n+1) = A163511((A243071(2n+1)-1) / 2).
a(n) = A249818(A250470(A249817(n))). [A250470 is an analogous operation for "going one step up" in the square array A083221 (A083140).]
(End)
Product_{k=1..n} a(k) = n! / A307035(n). - Vaclav Kotesovec, Mar 21 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.220703928... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Nov 18 2022

A245611 Permutation of natural numbers: a(n) = A243071(A064216(n)).

Original entry on oeis.org

0, 1, 3, 7, 2, 15, 31, 6, 63, 127, 14, 255, 5, 4, 511, 1023, 30, 13, 2047, 62, 4095, 8191, 12, 16383, 11, 126, 32767, 29, 254, 65535, 131071, 28, 61, 262143, 510, 524287, 1048575, 10, 27, 2097151, 8, 4194303, 125, 1022, 8388607, 59, 2046, 253, 16777215, 60, 33554431, 67108863, 26

Offset: 1

Antti Karttunen, Jul 28 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

The odd bisection of A243071 decremented by one and halved. (For a(1) = 0, take ceiling of -1/2).

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A245612.

Cf. A054429, A064216, A243071, A243065-A243066, A243505-A243506, A245607, A245609, A244153, A244154.

(define (A245611 n) (A243071 (A064216 n))) ;; offset 1, a(1) = 0.

a(1) = 0, and for n > 1, a(n) = (1/2) * (A243071((2*n)-1) - 1).
As a composition of related permutations:
a(n) = A243071(A064216(n)).
a(n) = A054429(A244153(n)).

A244153 Permutation of natural numbers, the odd bisection of A156552 halved; equally, a composition of A064216 and A156552: a(n) = A156552(A064216(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 16, 5, 32, 64, 9, 128, 6, 7, 256, 512, 17, 10, 1024, 33, 2048, 4096, 11, 8192, 12, 65, 16384, 18, 129, 32768, 65536, 19, 34, 131072, 257, 262144, 524288, 13, 20, 1048576, 15, 2097152, 66, 513, 4194304, 36, 1025, 130, 8388608, 35, 16777216, 33554432, 21, 67108864, 134217728, 2049, 268435456, 258, 67, 68, 24, 4097, 14

Offset: 1

Antti Karttunen, Jun 27 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A244154.

Cf. A054429, A064216, A156552, A243065-A243066, A243505-A243506, A245607, A245609, A245611.

(define (A244153 n) (A156552 (A064216 n)))

a(n) = A156552(2n+1) / 2.
As a composition of related permutations:
a(n) = A156552(A064216(n)).
a(n) = A054429(A245611(n)).

A245605 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 13, 18, 17, 26, 11, 12, 37, 34, 25, 74, 15, 16, 69, 50, 21, 14, 19, 20, 33, 138, 41, 66, 35, 52, 53, 22, 277, 82, 31, 32, 45, 554, 65, 90, 27, 36, 1109, 130, 101, 42, 43, 28, 73, 2218, 149, 30, 71, 104, 57, 146, 209, 114, 51, 148, 133, 70, 293, 418, 555, 164, 141, 586, 329, 282, 75, 68, 105, 106, 1173, 658, 23, 24

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

The even bisection halved gives A245607. The odd bisection incremented by one and halved gives A245707.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245606.
Cf. A048673, A064216, A064989, A245607, A245705, A245707.
Similar entanglement permutations: A244319, A005940, A163511, A243287, A243343, A243345, A244321, A245603, A245613, A135141, A237427.

A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A245605(n) = if(1==n, 1, if(0==(n%2), 2*A245605(A064989(n-1)), 1+(2*A245605(A064989(n)-1))));
for(n=1, 10001, write("b245605.txt", n, " ", A245605(n)));

;; With memoization-macro definec.
(definec (A245605 n) (cond ((= 1 n) 1) ((even? n) (* 2 (A245605 (A064989 (- n 1))))) (else (+ 1 (* 2 (A245605 (-1+ (A064989 n))))))))

a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).
a(1) = 1, a(2n) = 2 * a(A064216(n)), a(2n-1) = 1 + (2 * a(A064216(n)-1)).
As a composition of related permutations:
a(n) = A245607(A048673(n)).

A245608 Permutation of natural numbers, the even bisection of A245606 halved: a(n) = A245606(2*n)/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 13, 11, 6, 14, 18, 41, 7, 26, 28, 20, 9, 23, 63, 50, 25, 113, 313, 65, 12, 17, 88, 77, 172, 149, 43, 95, 16, 38, 33, 44, 10, 413, 163, 221, 19, 74, 48, 191, 22, 476, 118, 179, 49, 68, 138, 29, 39, 527, 78, 215, 31, 635, 1593, 227, 102, 71, 688, 242, 24, 122, 193, 104, 15, 98, 58, 176, 30, 32, 123

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245607.

Cf. A003961, A048673, A245606, A245609-A245610, A245706, A245708, A245709, A245612, A244154, A243065-A243066.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A245606(n) = if(1==n, 1, if(0==(n%2), 1+A003961(A245606(n/2)),A003961(1+A245606((n-1)/2))))
A245608(n) = A245606(2*n)/2;
for(n=1, 10001, write("b245608.txt", n, " ", A245608(n)))

(define (A245608 n) (A048673 (A245606 n)))

a(n) = A245606(2*n)/2.
As a composition of related permutations:
a(n) = A048673(A245606(n)).
a(n) = A245708(A245706(n)).
Other identities:
For all n >= 0, a(2^n) = A245708(2^n). Moreover, A245709 gives all such k that a(k) = A245708(k).

A243501 Permutation of even numbers: a(n) = 2*A048673(n).

Original entry on oeis.org

2, 4, 6, 10, 8, 16, 12, 28, 26, 22, 14, 46, 18, 34, 36, 82, 20, 76, 24, 64, 56, 40, 30, 136, 50, 52, 126, 100, 32, 106, 38, 244, 66, 58, 78, 226, 42, 70, 86, 190, 44, 166, 48, 118, 176, 88, 54, 406, 122, 148, 96, 154, 60, 376, 92, 298, 116, 94, 62, 316, 68, 112, 276, 730, 120

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

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

Cf. A243502, A245447, A245448, A244319, A245605-A245606, A245607-A245608.

a(n) = 2*A048673(n).
a(n) = A003961(n) + 1.
a(n) = A243502(A245447(n)).

A245610 Permutation of natural numbers: a(n) = A048673(A244319(n)).

Original entry on oeis.org

1, 3, 2, 13, 8, 4, 26, 7, 5, 28, 14, 172, 149, 25, 41, 18, 635, 102, 113, 1194, 11, 43, 428, 22, 17, 6, 77, 88, 71, 259, 527, 130, 227, 48, 74, 12, 677, 235, 20, 688, 68, 634, 5711, 61, 50, 1593, 1490, 27612, 59, 39, 29, 63, 11438, 10119, 4748, 9, 344, 238, 413, 1602, 941, 69

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245609.

Cf. A003961, A048673, A244319, A245607-A245608.

A048673(n) = (1+A003961(n))/2;
A245610(n) = A048673(A244319(n)); \\ For the other needed functions, see under A244319.
for(n=1, 10001, write("b245610.txt", n, " ", A245610(n)));

(define (A245610 n) (A048673 (A244319 n)))

a(n) = A048673(A244319(n)).

A245609 Permutation of natural numbers: a(n) = A244319(A064216(n)).

Original entry on oeis.org

1, 3, 2, 6, 9, 26, 8, 5, 56, 344, 21, 36, 4, 11, 204, 86, 25, 16, 176, 39, 518, 24, 125, 1376, 14, 7, 1268, 10, 51, 3186, 126, 1015, 298, 476, 305, 3204, 590, 115, 50, 5636, 15, 7118, 22, 825, 162, 2388, 153, 34, 626, 45, 4356, 144, 301, 156, 4374, 131, 816, 454, 49, 260, 44, 995, 52, 168, 81

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245610.

Cf. A064216, A064989, A244319, A245607-A245608.

A064216(n) = A064989((2*n)-1);
A245609(n) = A244319(A064216(n)); \\ For the other needed functions, see A244319.
for(n=1, 10001, write("b245609.txt", n, " ", A245609(n)));

(define (A245609 n) (A244319 (A064216 n)))

a(n) = A244319(A064216(n)).

A245603 Permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2a(k), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = 1+(2a(k)).

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 6, 16, 9, 7, 10, 12, 32, 17, 11, 13, 18, 14, 20, 24, 33, 19, 64, 15, 21, 25, 34, 22, 26, 36, 28, 40, 65, 35, 23, 27, 48, 37, 29, 41, 66, 38, 128, 30, 42, 49, 50, 68, 67, 44, 39, 52, 72, 129, 31, 43, 51, 69, 56, 45, 80, 53, 130, 73, 57, 70, 46, 54, 81, 96, 74, 58, 82, 131, 132, 76, 71, 256, 60

Offset: 1

Antti Karttunen, Jul 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245604.
Cf. A026424, A028260, A066829, A055037, A055038.
Similar permutations: A143692, A244152, A244321, A245613, A245605, A245607.

a(1) = 1, and for n > 1, if A066829(n) = 1, then a(n) = 2 * A245603(A055038(n)), otherwise a(n) = 1 + (2 * A245603(A055037(n)-1)).
As a composition of related permutations:
a(n) = A244321(A245613(n)).
For all n >= 1, A000035(a(n)) = 1 - A066829(n). [Permutation A143692 has the same property.]

cp's OEIS Frontend

A245705 Permutation of natural numbers: a(n) = A245607(A245708(n)).

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A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A245611 Permutation of natural numbers: a(n) = A243071(A064216(n)).

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A244153 Permutation of natural numbers, the odd bisection of A156552 halved; equally, a composition of A064216 and A156552: a(n) = A156552(A064216(n)).

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A245605 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

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A245608 Permutation of natural numbers, the even bisection of A245606 halved: a(n) = A245606(2*n)/2.

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A243501 Permutation of even numbers: a(n) = 2*A048673(n).

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A245610 Permutation of natural numbers: a(n) = A048673(A244319(n)).

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