A064216 -id:A064216 - cp's OEIS frontend

A245448 Permutation of natural numbers: a(n) = A064216(A064216(n)).

1, 2, 3, 4, 5, 11, 10, 7, 9, 14, 17, 31, 13, 6, 12, 34, 8, 23, 59, 41, 71, 16, 19, 39, 25, 26, 58, 37, 61, 30, 44, 22, 33, 49, 18, 85, 86, 15, 38, 69, 29, 151, 35, 55, 42, 107, 57, 97, 106, 21, 191, 122, 53, 111, 134, 74, 145, 109, 46, 82, 89, 50, 47, 36, 157, 133, 121, 43, 92, 110, 68, 52, 131, 28

Offset: 1

Antti Karttunen, Jul 22 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: A245447.
Fixed points: A245449.
Cf. A064216, A243501, A243502.

(define (A245448 n) (A064216 (A064216 n)))

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

For all n >= 1, A243502(n) = A243501(a(n)).

A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

Original entry on oeis.org

1, 0, 0, 0, 0, -1, 5, -8, 0, 6, 3, -2, 0, -19, 5, 4, 4, -20, 19, -22, -6, 15, -3, 8, 0, 0, -16, -16, 18, -24, 40, -70, -9, 24, -21, 8, 50, -55, -8, 24, -6, 31, 15, -58, -20, 17, 31, -92, -2, -70, 37, 24, 0, 20, 49, 18, -6, -26, 13, -33, 15, -62, -158, -20, 22, -15, 49, -130, 67, 48, 49, -58, 29, -112, -4, 60, -73, -16

Offset: 1

Antti Karttunen, Nov 19 2021

sign

Dirichlet convolution of A064216 with A323893, which is the Dirichlet inverse of A048673. Therefore, convolving A048673 with this sequence gives A064216.

Note how for n = 1 .. 35, a(n) = -A349398(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A048673, A064216, A064989, A323893, A349398 (Dirichlet inverse), A349399 (sum with it), A349384.

Cf. also pairs A349376, A349377 and A349613, A349614 for similar constructions.

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A064216(n) = { my(f = factor(n+n-1)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349397(n) = sumdiv(n,d,A064216(n/d)*A323893(d));

a(n) = Sum_{d|n} A064216(n/d) * A323893(d).

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)).

A286372 a(n) = A286366(A064216(n)).

Original entry on oeis.org

4, 6, 8, 13, 4, 9, 8, 11, 13, 12, 14, 8, 28, 6, 9, 13, 11, 21, 9, 11, 13, 12, 8, 8, 40, 14, 9, 65, 14, 13, 8, 13, 64, 13, 11, 8, 9, 30, 20, 12, 4, 9, 21, 11, 8, 21, 14, 20, 12, 9, 12, 13, 23, 9, 8, 11, 13, 64, 8, 84, 28, 14, 116, 12, 14, 9, 85, 11, 8, 12, 11, 65, 65, 42, 8, 13, 13, 21, 9, 11, 21, 13, 66, 8, 28, 12, 9, 49, 14, 13, 8, 11, 65, 20, 14, 13, 9, 66

Offset: 1

Antti Karttunen, May 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064216, A278223, A286243, A286366, A286371, A286373.

(define (A286372 n) (A286366 (A064216 n)))

a(n) = A286366(A064216(n)).

A243502 Permutation of even numbers: a(n) = 2 * A064216(n).

Original entry on oeis.org

2, 4, 6, 10, 8, 14, 22, 12, 26, 34, 20, 38, 18, 16, 46, 58, 28, 30, 62, 44, 74, 82, 24, 86, 50, 52, 94, 42, 68, 106, 118, 40, 66, 122, 76, 134, 142, 36, 70, 146, 32, 158, 78, 92, 166, 110, 116, 102, 178, 56, 194, 202, 60, 206, 214, 124, 218, 114, 88, 130, 98, 148, 54, 226, 164, 254, 170, 48

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

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

Cf. A005843, A064216, A243500, A243501, A245447, A245448.

Scheme
```
(define (A243502 n) (* 2 (A064216 n)))
```

a(n) = 2 * A064216(n) = A005843(A064216(n)).

a(n) = A243501(A245448(n)).

A246268 Permutation of natural numbers: a(n) = A246264(A064216(2*n)).

Original entry on oeis.org

1, 2, 4, 3, 7, 9, 5, 16, 6, 12, 18, 19, 14, 11, 24, 10, 29, 34, 8, 38, 43, 21, 26, 23, 15, 52, 53, 30, 27, 32, 39, 56, 62, 13, 66, 41, 22, 69, 46, 49, 73, 77, 35, 20, 84, 59, 48, 88, 33, 98, 57, 40, 100, 60, 68, 106, 116, 36, 64, 119, 17, 93, 125, 42, 72, 132, 80, 140, 31, 58, 145, 91, 86, 74, 104, 96, 151, 158, 28

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

This permutation is induced when A064216 is restricted to even numbers (equally, when A064989 is restricted to the numbers of form 4n-1) and the resulting numbers are "ranked" with A246264.

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

Inverse: A246267.

Cf. also A064216, A064989, A246264, A246266.

a(n) = A246264(A064216(2*n)).

a(n) = A246264(A064989((4*n)-1)).

A246364 Permutation of natural numbers: a(n) = A064216(A227413(n)).

Original entry on oeis.org

1, 2, 5, 3, 7, 11, 13, 4, 6, 9, 19, 14, 8, 12, 29, 10, 17, 31, 23, 16, 41, 71, 37, 44, 47, 39, 43, 42, 38, 30, 26, 59, 22, 34, 15, 85, 53, 58, 25, 130, 57, 151, 61, 311, 103, 69, 33, 365, 157, 111, 73, 226, 74, 106, 67, 370, 223, 56, 97, 341, 139, 122, 35, 133, 55, 86, 20, 145, 46, 49, 21, 659, 118, 36, 83, 419, 127, 191, 18

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

After a(2) = 2, the rest of the even bisection contains only terms of A246261. However, some of the terms of A246261 are also found in the odd bisection, while terms of A246263, apart from 2, all reside in the odd bisection of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A246363.
Related or similar permutations: A064216, A227413, A246366, A246368.
Cf. A246261, A246263.

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
A227413(n) = if(1==n, 1, if(!(n%2), prime(A227413(n/2)), A002808(A227413((n-1)/2))));
A246364(n) = A064216(A227413(n));
for(n=1, 4095, write("b246364.txt", n, " ", A246364(n)));

(define (A246364 n) (A064216 (A227413 n)))

a(n) = A064216(A227413(n)).

A270430 Numbers n such that A048673(n) and A064216(n) are of the same parity.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 16, 17, 20, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 48, 49, 50, 52, 53, 58, 62, 64, 65, 68, 69, 74, 75, 77, 80, 81, 82, 85, 90, 93, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 109, 111, 113, 114, 116, 117, 120, 121, 124, 125, 126, 128, 130, 132, 133, 136, 137, 139, 141, 144

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

See A270434 for the possible bias favoring this sequence over the complement A270431.

Antti Karttunen, Table of n, a(n) for n = 1..17000

Complement: A270431.
Left inverse: A270432.
Cf. A245449 (a subsequence).
Cf. A000035, A048673, A064216, A270434.
Cf. also A269860.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; Select[Range@ 144, Xor[EvenQ@ f@ #, OddQ@ g@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

Other identities. For all n >= 1:

A270432(a(n)) = n.

A270431 Numbers n such that A048673(n) and A064216(n) are of opposite parity.

Original entry on oeis.org

6, 7, 11, 14, 15, 18, 19, 21, 22, 23, 24, 28, 35, 38, 43, 44, 45, 46, 47, 51, 54, 55, 56, 57, 59, 60, 61, 63, 66, 67, 70, 71, 72, 73, 76, 78, 79, 83, 84, 86, 87, 88, 89, 91, 92, 94, 95, 96, 103, 107, 110, 112, 115, 118, 119, 122, 123, 127, 129, 131, 134, 135, 138, 140, 142, 143, 146, 150, 152, 153, 157, 158, 159, 162

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

See comments in A270434.

Antti Karttunen, Table of n, a(n) for n = 1..17000

Complement: A270430.
Left inverse: A270433.
Cf. A000035, A048673, A064216, A270434.
Cf. also A269861.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; Select[Range@ 162, Xor[EvenQ@ f@ #, EvenQ@ g@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

Other identities. For all n >= 1:

A270433(a(n)) = n.

A285703 a(n) = A000203(A064216(n)).

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 12, 14, 18, 18, 20, 13, 15, 24, 30, 24, 24, 32, 36, 38, 42, 28, 44, 31, 42, 48, 32, 54, 54, 60, 42, 48, 62, 60, 68, 72, 39, 48, 74, 31, 80, 56, 72, 84, 72, 90, 72, 90, 56, 98, 102, 72, 104, 108, 96, 110, 80, 84, 84, 57, 114, 40, 114, 126, 128, 108, 60, 132, 138, 132, 96, 96, 93, 140, 150, 98, 120, 152, 144, 120, 158, 96, 164, 133, 126

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A064216, A099774, A151799, A285702, A285704, A285705.

Table[DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 86}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285703 n) (A000203 (A064216 n)))

a(n) = A000203(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.8168476756..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

cp's OEIS Frontend

A245448 Permutation of natural numbers: a(n) = A064216(A064216(n)).

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A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

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

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A286372 a(n) = A286366(A064216(n)).

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A243502 Permutation of even numbers: a(n) = 2 * A064216(n).

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A246268 Permutation of natural numbers: a(n) = A246264(A064216(2*n)).

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A246364 Permutation of natural numbers: a(n) = A064216(A227413(n)).

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A270430 Numbers n such that A048673(n) and A064216(n) are of the same parity.

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Mathematica

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A270431 Numbers n such that A048673(n) and A064216(n) are of opposite parity.

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