A250469 -id:A250469 - cp's OEIS frontend

A269867 Self-inverse permutation of natural numbers: a(1) = 1, for even n, a(n) = A250469(1+a(n/2)), for odd n, a(n) = 2*a(A268674(n)-1).

1, 3, 2, 9, 6, 5, 18, 27, 4, 11, 10, 15, 22, 23, 12, 81, 30, 7, 162, 33, 36, 13, 14, 45, 54, 29, 8, 69, 26, 17, 138, 243, 20, 37, 46, 21, 34, 167, 44, 99, 42, 41, 198, 39, 24, 35, 82, 135, 90, 91, 60, 87, 70, 25, 66, 207, 324, 65, 174, 51, 130, 149, 72, 729, 58, 55, 102, 111, 28, 53, 110, 63, 106, 77, 108, 501, 74, 115, 126, 297, 16, 47

Offset: 1

Antti Karttunen, Mar 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3165
Index entries for sequences that are permutations of the natural numbers

Cf. A250469, A268674.

Similar permutations: A269865, A269866, A270197.

a(1) = 1, a(2n) = A250469(1+a(n)), a(2n+1) = 2*a(A268674(2n+1)-1).

A266404 Self-inverse permutation of natural numbers: a(n) = A250470(A030101(A250469(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 17, 10, 7, 12, 13, 20, 25, 16, 9, 18, 23, 14, 53, 22, 19, 28, 15, 36, 27, 24, 29, 40, 37, 32, 33, 34, 83, 26, 31, 42, 51, 30, 47, 38, 59, 44, 101, 76, 41, 60, 73, 68, 39, 52, 21, 84, 107, 56, 131, 72, 43, 48, 89, 80, 125, 64, 65, 66, 109, 50, 99, 82, 71, 58, 49, 74, 151, 46, 239, 78, 97, 62, 173, 70, 35, 54

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A030101, A057889, A250469, A250470.
Cf. A265329, A266402 (other conjugates or similar derivations of A057889).
Cf. also A266403.

(define (A266404 n) (A250470 (A030101 (A250469 n))))

a(n) = A250470(A030101(A250469(n))) = A250470(A057889(A250469(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 9, 18, 17, 20, 31, 22, 25, 24, 21, 26, 27, 28, 19, 30, 29, 32, 49, 34, 71, 36, 37, 38, 39, 40, 41, 42, 43, 44, 107, 46, 47, 48, 119, 50, 51, 52, 35, 54, 89, 56, 101, 58, 53, 60, 61, 62, 63, 64, 115, 66, 67, 68, 173, 70, 55, 72, 33, 74, 75, 76, 131, 78, 77, 80, 167

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

			For n=9 we first find what number is below 9 in square array A083221, which is 25, then we find what number is above 25 in square array A255127, which is 23, thus a(9) = 23.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A269356.
Cf. A250469, A269380.
Cf. also arrays A083221 & A255127.
More recursed variant: A269357. Cf. also permutations A266645, A255407, A269171.

(define (A269355 n) (A269380 (A250469 n)))

a(n) = A269380(A250469(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes the even numbers.]

A269357 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269380(A250469(2n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 9, 46, 17, 20, 31, 22, 25, 24, 21, 26, 27, 28, 19, 30, 29, 32, 49, 18, 71, 92, 37, 34, 39, 40, 41, 62, 43, 44, 107, 50, 47, 48, 119, 42, 51, 52, 35, 54, 89, 56, 101, 38, 53, 60, 61, 58, 63, 64, 115, 98, 67, 36, 173, 142, 55, 184, 33, 74, 75, 68, 131, 78, 77, 80, 167

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

This is a variant of A269355, from which it differs for the first time at n=18.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A269358.

Cf. A000035, A250469, A269380, A269355, A269172.

a(1) = 1, after which, for even n, a(n) = 2*a(n/2) and for odd n, a(n) = A269355(n) = A269380(A250469(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A269863 Permutation of natural numbers: a(1) = 1, a(A269360(n)) = 2a(n), a(A250469(1+n)) = 1 + 2a(n).

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, 19, 20, 69, 50, 21, 14, 15, 16, 41, 138, 33, 82, 27, 36, 53, 22, 277, 66, 35, 52, 45, 554, 105, 90, 23, 24, 1109, 210, 101, 42, 75, 68, 49, 2218, 149, 38, 51, 148, 137, 98, 297, 274, 39, 40, 29, 30, 197, 594, 139, 100, 61, 394, 201, 122, 43, 28, 73, 106, 789, 402, 31, 32

Offset: 1

Antti Karttunen, Mar 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A269864.
Cf. A250469, A268674, A269360.
Differs from similarly constructed A245605 for the first time at n=21, where a(21)=19, instead of 15.

a(1) = 1, after which for even n, a(n) = 2*a(A268674(n-1)), for odd n, a(n) = 1 + 2*a(A268674(n)-1).

A269864 Permutation of natural numbers: a(1) = 1, a(2n) = A269360(a(n)), a(2n+1) = A250469(1+a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 15, 16, 11, 26, 27, 28, 13, 12, 21, 22, 25, 36, 45, 46, 19, 14, 33, 76, 65, 66, 81, 82, 31, 18, 39, 34, 17, 56, 63, 64, 29, 50, 75, 106, 41, 116, 135, 136, 53, 24, 57, 40, 35, 86, 99, 226, 143, 120, 195, 196, 71, 206, 243, 244, 89, 38, 93, 52, 23, 96, 117, 100, 77, 20, 51, 166, 145, 156, 189, 190, 119, 32, 87

Offset: 1

Antti Karttunen, Mar 13 2016

This sequence can be represented as a binary tree. When the parent contains n, the left hand child contains A269360(n) [= 1+A250469(n)] and the right hand child contains A250469(1+n):
                                     1
                                     |
                  ................../ \..................
                 2                                       3
       4......../ \........5                   6......../ \........9
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  10       7           8       15         16       11         26       27
28  13   12 21       22 25   36  45     46  19   14  33     76  65   66  81
etc.

Antti Karttunen, Table of n, a(n) for n = 1..5887
Index entries for sequences that are permutations of the natural numbers

Inverse: A269863.
Cf. A250469, A269360.
Cf. also A269865, A269866, A269867.
Differs from similarly constructed A245606 for the first time at n=15, where a(15)=27 instead of 21.

a(1) = 1, a(2n) = 1 + A250469(a(n)), a(2n+1) = A250469(1+a(n)).

A280702 a(n) = gcd(A003961(n), A250469(n)).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 3, 25, 3, 13, 3, 17, 3, 35, 9, 19, 3, 23, 3, 55, 3, 29, 3, 49, 3, 5, 9, 31, 3, 37, 3, 5, 3, 77, 15, 41, 3, 5, 9, 43, 3, 47, 3, 5, 3, 53, 3, 121, 147, 5, 153, 59, 3, 91, 33, 5, 3, 61, 3, 67, 3, 5, 27, 119, 195, 71, 3, 5, 3, 73, 3, 79, 3, 5, 9, 143, 3, 83, 3, 5, 3, 89, 3, 133, 3, 5, 9, 97, 3, 187, 3, 5, 3, 161

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

For n > 1, a(n) > 1 because A020639(A003961(n)) = A020639(A250469(n)) = A003961(A020639(n)).

Antti Karttunen, Table of n, a(n) for n = 1..13330

Cf. A003961, A020639, A250469, A280497, A280498, A280692, A280703, A280704, A283463, A283464.

f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ GCD[ Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1], f@ First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017 *)

(define (A280702 n) (gcd (A003961 n) (A250469 n)))

a(n) = gcd(A003961(n), A250469(n)).

A346479 Dirichlet inverse of A250469.

Original entry on oeis.org

1, -3, -5, 0, -7, 15, -11, 6, 0, 15, -13, 12, -17, 27, 35, 0, -19, 24, -23, 42, 55, 15, -29, -66, 0, 27, 60, 54, -31, -27, -37, -12, 45, 15, 77, -144, -41, 27, 75, -102, -43, -63, -47, 132, 60, 39, -53, -24, 0, 84, 65, 144, -59, -384, 91, -162, 85, 15, -61, -558, -67, 39, 120, 0, 119, 165, -71, 222, 115, 9, -73, 168

Offset: 1

Antti Karttunen, Jul 30 2021

sign

Not all zeros occur on squares. For example, a(1445) = a(5 * 17^2) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A250469, A346480.

Cf. also A346234, A346477.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA250469(n/d).

a(n) = A346480(n) - A250469(n).

A269360 Permutation of even numbers: a(n) = 1 + A250469(n).

Original entry on oeis.org

2, 4, 6, 10, 8, 16, 12, 22, 26, 28, 14, 34, 18, 40, 36, 46, 20, 52, 24, 58, 56, 64, 30, 70, 50, 76, 66, 82, 32, 88, 38, 94, 86, 100, 78, 106, 42, 112, 96, 118, 44, 124, 48, 130, 116, 136, 54, 142, 122, 148, 126, 154, 60, 160, 92, 166, 146, 172, 62, 178, 68, 184, 156, 190, 120, 196, 72, 202, 176, 208, 74, 214, 80, 220, 186, 226, 144

Offset: 1

Antti Karttunen, Mar 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A253886, A250469, A243501.

Cf. also A269863, A269864.

(* b = A250469 *) b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[1, 1]]; For[ k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[ FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]];
a[n_] := b[n] + 1;
Array[a, 100] (* Jean-François Alcover, Mar 14 2016 *)

Scheme
```
(define (A269360 n) (+ 1 (A250469 n)))
```

a(n) = 1 + A250469(n).

a(n) = 2 + A253886(n-1).

A280693 Numbers n such that A003961(n) = A250469(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 50, 52, 53, 55, 59, 61, 65, 66, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 186, 187, 191, 193, 197, 199, 203

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

Positions of zeros in A280692. Conjectured to be also the positions of ones in A280703.
For most terms k of this sequence A003961(k) is also included as a term. Exceptions are: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 11605, ... that seems to be a subsequence of all those terms that have more than two prime factors: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 8695, 11605, ...
Note how 8695 = 5*37*47 and A003961(8695) = 7*41*53 = 15211 = A003961(8695) = A250469(8695) (for no apparent reason?).

Antti Karttunen, Table of n, a(n) for n = 1..2001

Fixed points of permutations A266645 & A266646.
Cf. A001222, A003961, A250469, A280692, A280703.
Cf. A000040, A001248, A006094, A251728 (subsequences).
Cf. also arrays A083221 and A246278.

f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; With[{nn = 204}, Function[s, Function[t, Select[Range@ nn, f@ # == t[[#]] &]]@ MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ nn]]@ PositionIndex@ Array[g, 10^4]] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

cp's OEIS Frontend

A269867 Self-inverse permutation of natural numbers: a(1) = 1, for even n, a(n) = A250469(1+a(n/2)), for odd n, a(n) = 2*a(A268674(n)-1).

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A266404 Self-inverse permutation of natural numbers: a(n) = A250470(A030101(A250469(n))).

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A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

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A269357 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269380(A250469(2n+1)).

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A269863 Permutation of natural numbers: a(1) = 1, a(A269360(n)) = 2*a(n), a(A250469(1+n)) = 1 + 2*a(n).

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A269864 Permutation of natural numbers: a(1) = 1, a(2n) = A269360(a(n)), a(2n+1) = A250469(1+a(n)).

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A280702 a(n) = gcd(A003961(n), A250469(n)).

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A346479 Dirichlet inverse of A250469.

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PARI

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A269360 Permutation of even numbers: a(n) = 1 + A250469(n).

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A269863 Permutation of natural numbers: a(1) = 1, a(A269360(n)) = 2a(n), a(A250469(1+n)) = 1 + 2a(n).