A234017 -id:A234017 - cp's OEIS frontend

A213714 Inverse function for injection A005187.

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

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

a(0)=0; thereafter if n occurs as a term of A005187, a(n)=its position in A005187, otherwise zero. This works as an "inverse" function for A005187 in a sense that a(A005187(n)) = n for all n.

a(n)*A234017(n) = 0 for all n.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Can be used when computing A213715, A213723, A213724, A233275, A233277. Cf. A005187, A046699, A079559, A234017, A230414.

import Data.List (genericIndex)
a213714 n = genericIndex a213714_list n
a213714_list = f [0..] a005187_list 0 where
   f (x:xs) ys'@(y:ys) i | x == y    = i : f xs ys (i+1)
                         | otherwise = 0 : f xs ys' i
-- Reinhard Zumkeller, May 01 2015

from sympy import factorial
def a046699(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1))>0: s+=1
    return s
def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
def a043545(n):
    x=bin(n)[2:]
    return int(max(x)) - int(min(x))
def a079559(n): return 1 if n==0 else a043545(n + 1)*a079559(n + 1 - a053644(n + 1))
def a(n): return 0 if n==0 else a079559(n)*(a046699(n + 2) - 1) # Indranil Ghosh, Jun 11 2017

a(0)=0, for n>0, a(n) = A079559(n) * (A046699(n+2)-1) [With A046699's October 2012 starting offset. Incorrect indexing shown in this formula corrected by Antti Karttunen, Dec 18 2013]

A233275 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with even and odd numbers.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 14, 10, 15, 11, 9, 8, 24, 25, 26, 28, 27, 29, 20, 30, 21, 22, 18, 31, 23, 19, 17, 16, 48, 49, 50, 52, 51, 53, 56, 54, 57, 58, 40, 55, 59, 41, 60, 42, 61, 44, 36, 43, 45, 62, 46, 37, 38, 34, 63, 47, 39, 35, 33, 32, 96, 97, 98, 100

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

It seems that for all n, a(A000079(n)) = A003945(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python Program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233276.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233277/A233278, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=1, a(n) = 2*a(A213714(n)-1), else a(n) = 1+(2*a(A234017(n))).

a(n) = A054429(A233277(n)). [Follows from the definitions of these sequences]

A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 9, 13, 8, 12, 14, 15, 23, 22, 21, 19, 20, 18, 27, 17, 26, 25, 29, 16, 24, 28, 30, 31, 47, 46, 45, 43, 44, 42, 39, 41, 38, 37, 55, 40, 36, 54, 35, 53, 34, 51, 59, 52, 50, 33, 49, 58, 57, 61, 32, 48, 56, 60, 62, 63, 95, 94, 93, 91

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Indranil Ghosh, PARI program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233278.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=0, a(n) = 2*a(A234017(n)), else a(n) = 1+(2*a(A213714(n)-1)).

a(n) = A054429(A233275(n)). [Follows from the definitions of these sequences]

A255560 One-based row index of n in array A255555.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Equally: One-based column index of n in array A255557.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A005187, A055938, A213714, A234017, A255555, A255557.
Cf. also A255559 (corresponding column index).
Cf. A254112, A256990.

a(1) = 1; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1+A234017(n), otherwise a(n) = a(A213714(n)-1).

In other words, a(1) = 1, and for n > 1, if n = A055938(k) for some k, then a(n) = k+1, otherwise it must be that n = A005187(h) for some h, in which case a(n) = a(h-1).

A256990 One-based row index of n in array A256995.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Also one-based column index for array A256997.

a(1) = 0 by convention, as 1 is outside of the actual arrays A256995 & A256997.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A005187, A055938, A213714, A234017, A256995, A256997.

Cf. A256989 (corresponding column index), A255560.

a(1) = 0; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = A234017(n), otherwise a(n) = a(A213714(n)).

In other words, a(1) = 0, and for n > 1, if n = A055938(k) for some k, then a(n) = k, otherwise it must be that n = A005187(h) for some h, in which case a(n) = a(h).

A234016 Partial sums of the characteristic function of A055938.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 13, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 19, 20, 21, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 26, 26, 26, 27, 28, 29, 30, 31, 31, 31, 32, 32, 32, 33, 34

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Also: a(0) = a(1) = 0, and thereafter, a(n) = the largest k such that A055938(k) <= n.

Conjecture: partial sums of A308187 (i.e, A308187 is the characteristic function of A055938). - Sean A. Irvine, Jul 16 2022

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A046699, A055938, A079559, A234017, A233275, A308187.

from sympy import factorial
def a046699(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1)): s+=1
    return s
def a(n): return n - (a046699(n + 2) - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

If n < 2, a(n)=0, otherwise a(n) = a(n-1) + (1-A079559(n)).

a(n) = n - (A046699(n+2)-1) [With A046699's October 2012 starting offset].

A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 13, 14, 12, 11, 10, 9, 8, 31, 28, 29, 30, 23, 25, 27, 26, 22, 24, 21, 20, 19, 18, 17, 16, 63, 59, 60, 61, 50, 52, 62, 53, 55, 56, 58, 41, 44, 49, 57, 51, 46, 54, 48, 40, 43, 47, 45, 39, 42, 38, 37, 36, 35, 34, 33, 32, 127, 122, 123, 124, 108, 110, 125, 111, 113, 114, 126, 89, 92, 117, 115, 118, 94, 119, 121

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276343.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276346, A276442.

(definec (A276344 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A087686 (+ 1 (A276344 (- (A213714 n) 1))))) (else (A088359 (A276344 (A234017 n))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A087686(1+a(A213714(n)-1)), otherwise a(n) = A088359(a(A234017(n))).
As a composition of other permutations:
a(n) = A267112(A233275(n)).
a(n) = A276442(A233277(n)).

A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 12, 9, 13, 8, 15, 14, 11, 19, 24, 22, 18, 27, 21, 23, 17, 29, 28, 25, 16, 31, 30, 26, 20, 36, 45, 43, 40, 54, 51, 35, 49, 42, 39, 41, 58, 48, 53, 34, 52, 38, 50, 44, 61, 60, 33, 59, 56, 55, 46, 32, 63, 62, 57, 47, 37, 69, 83, 81, 78, 102, 99, 74, 97, 93, 91, 68, 116, 112, 80, 88, 77, 109, 73, 75, 96, 90

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276345.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276344, A276442.

(definec (A276346 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A088359 (A276346 (- (A213714 n) 1)))) (else (A087686 (+ 1 (A276346 (A234017 n)))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A088359(a(A213714(n)-1)), otherwise a(n) = A087686(1+a(A234017(n))).
As a composition of other permutations:
a(n) = A276442(A233275(n)).
a(n) = A267112(A233277(n)).

cp's OEIS Frontend

A213714 Inverse function for injection A005187.

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A233275 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with even and odd numbers.

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A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

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A255560 One-based row index of n in array A255555.

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A256990 One-based row index of n in array A256995.

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A234016 Partial sums of the characteristic function of A055938.

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A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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