A213714 -id:A213714 - cp's OEIS frontend

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

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]

A234017 Inverse function for injection A055938.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

a(0)=0; thereafter if n occurs as a term of A055938, a(n)=its position in A055938, otherwise zero. This works as an "inverse" function for A055938 in a sense that a(A055938(n)) = n for all n.

a(n)*A213714(n) = 0 for all n.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence

Cf. also A079559, A234016, A055938, A213714, A233275, A233277.

(define (A234017 n) (* (- 1 (A079559 n)) (A234016 n)))

a(n) = (1-A079559(n)) * A234016(n).

A213723 a(n) = smallest natural number x such that x=n+A000120(x), otherwise zero.

Original entry on oeis.org

0, 2, 0, 4, 6, 0, 0, 8, 10, 0, 12, 14, 0, 0, 0, 16, 18, 0, 20, 22, 0, 0, 24, 26, 0, 28, 30, 0, 0, 0, 0, 32, 34, 0, 36, 38, 0, 0, 40, 42, 0, 44, 46, 0, 0, 0, 48, 50, 0, 52, 54, 0, 0, 56, 58, 0, 60, 62, 0, 0, 0, 0, 0, 64, 66, 0, 68, 70, 0, 0, 72, 74, 0, 76, 78

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

			a(1) = 2, as 2 is the smallest natural number such that x such that x=1+A000120(x) (as 2=1+A000120(2)=1+1).
a(2) = 0, as there are no solutions for 2, because it belongs to A055938.
a(11) = 14, as 14 is the smallest natural number x such that x=11+A000120(x) (as 14=11+A000120(14)=11+3).

Antti Karttunen, Table of n, a(n) for n = 0..1024

a(A055938(n)) = 0. a(A005187(n)) = A005843(n) = 2n.

Cf. A213724. Used for computing A213725-A213727. Cf. A179016.

a213723 = (* 2) . a213714  -- Reinhard Zumkeller, May 01 2015

(define (A213723 n) (A005843 (A213714 n)))

a(n) = 2*A213714(n).

Also, by partitioning into sums of distinct nonzero terms of A000225: if n can be formed as a sum of (2^a)-1 + (2^b)-1 + (2^c)-1, etc. where the exponents a, b, c are distinct and all > 0, then a(n) = 2^a + 2^b + 2^c, etc. If this is not possible, then n is one of the terms of A055938, and a(n)=0.

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]

A213710 Number of steps to reach 0 when starting from 2^n and iterating the map x -> x - (number of 1's in binary representation of x): a(n) = A071542(2^n) = A218600(n)+1.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 22, 39, 69, 123, 221, 400, 730, 1344, 2494, 4656, 8728, 16406, 30902, 58320, 110299, 209099, 397408, 757297, 1446946, 2771952, 5323983, 10250572, 19780123, 38243221, 74058514, 143592685, 278661809, 541110612, 1051158028, 2042539461, 3969857206

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

Conjecture: A179016(a(n))= 2^n for all n apart from n=2. This is true if all powers of 2 except 2 itself occur in A179016 as in that case they must occur at positions given by this sequence.

This is easy to prove: It suffices to note that after 3 no integer of form (2^k)+1 can occur in A005187, thus for all k >= 2, A213725((2^k)+1) = 1 or equally: A213714((2^k)+1) = 0. - Antti Karttunen, Jun 12 2013

One more than A218600, which is the partial sums of A213709, thus the latter also gives the first differences of this sequence.
Cf. A000079, A005187, A071542, A218616, A233271.
Analogous sequences: A219665, A255062.

a(n) = A071542(A000079(n)) = A071542(2^n).

a(n) = 1 + A218600(n).

a(29)-a(36) from Alois P. Heinz, Jul 03 2022

A255559 One-based column index of n in array A255555.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 3, 3, 1, 2, 2, 1, 1, 1, 4, 4, 1, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 5, 5, 1, 2, 3, 1, 1, 4, 2, 1, 2, 4, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 6, 6, 1, 2, 3, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 3, 2, 1, 3, 5, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 1, 4, 4, 1, 1, 2, 2, 1, 3, 3, 1

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

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

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

Cf. A005187, A055938, A070939, A213714, A255555, A255557.
Cf. also A255560 (corresponding row index).
Cf. A254111, A256989, A256478, A256993.

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, otherwise 1 + a(A213714(n)-1).
In other words, a(1) = 1, and for n > 1, if n = A005187(k) for some k, then a(n) = 1 + a(k-1), otherwise it must be that n is in A055938, in which case a(n) = 1.
Other identities and observations. For all n >= 1:
a(n) <= A256478(n) <= A070939(n).
a(n) <= A256993(n) + 1.

A256989 One-based column index of n in array A256995.

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 3, 2, 1, 3, 6, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 1, 4, 7, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 1, 1, 2, 6, 1, 3, 2, 1, 1, 2, 3, 1, 3, 2, 1, 1, 1, 3, 5, 1, 2, 2, 1, 1, 2, 2, 1, 2, 4

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Also one-based row 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, A256995, A256997.

Cf. A256990 (corresponding row index), A255559.

a(1) = 0; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1, otherwise a(n) = 1 + a(A213714(n)).
In other words, a(1) = 0, and for n > 1, if n = A005187(k) for some k, then a(n) = 1 + a(k), otherwise it must be that n is in A055938, in which case a(n) = 1.
Other observations. For all n >= 1 it holds that:
a(n) <= A256993(n).

A230414 Inverse function for injection A219650.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 31 2013

nonn

a(0)=0; thereafter if n occurs as a term of A219650, a(n)=its position in A219650, otherwise zero. This works as an "inverse" function for A219650 in a sense that a(A219650(n)) = n for all n.

Antti Karttunen, Table of n, a(n) for n = 0..10079

Cf. A219650, A230423, A230424. A219658 gives the positions of zeros after the first one. Analogous sequence for binary system: A213714.

(define (A230414 n) (* (A230412 n) (A230413 n)))

a(n) = A230412(n) * A230413(n).

a(n) = A230423(n)/2.

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).

cp's OEIS Frontend

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

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A234017 Inverse function for injection A055938.

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A213723 a(n) = smallest natural number x such that x=n+A000120(x), otherwise zero.

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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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A213710 Number of steps to reach 0 when starting from 2^n and iterating the map x -> x - (number of 1's in binary representation of x): a(n) = A071542(2^n) = A218600(n)+1.

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A255559 One-based column index of n in array A255555.

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A256989 One-based column index of n in array A256995.

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A230414 Inverse function for injection A219650.

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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.