A126760 -id:A126760

A249746

Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

+20
14

1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, 15, 16, 59, 18, 41, 32, 20, 31, 39, 21, 23, 92, 40, 24, 49, 25, 27, 82, 48, 28, 209, 30, 45, 52, 33, 63, 62, 54, 34, 109, 35, 36, 129, 37, 38, 69, 43, 68, 142, 70, 57, 72, 115, 44, 79, 46, 85, 292, 47, 50, 89, 74, 73, 202, 51, 53, 159, 87, 55, 99, 107, 56, 152, 58, 97, 192, 60

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Permutation obtained from the odd bisection of A003961 (or from the odd bisection of A048673).

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = 1 + f(A003961(2n - 1)), where f(n) = 2*floor[n/6] + ((n mod 6)-1)/4. [Here 1 + f(A007310(n)) = n.]

a(n) = A126760(A249735(n)). - Antti Karttunen, Jul 25 2016

As a composition of related permutations:

a(n) = A249824(A064216(n)).

Other identities. For all n >= 1:

A249735(n) = A007310(a(n)).

a(3n-1) = A273669(a(n)) and a(A254049(n)) = A273664(a(n)). - Antti Karttunen, Aug 07 2016

EXAMPLE

a(5) = 9 because of the following. 2*A064216(5) = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(5) = 9.

MATHEMATICA

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^6]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Map[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 #]] &, Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 87}]] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

PROG

(Scheme)

(define (A249746 n) (define (Ainv_of_A007310off0 n) (+ (* 2 (floor->exact (/ n 6))) (/ (- (modulo n 6) 1) 4))) (+ 1 (Ainv_of_A007310off0 (A003961 (+ n n -1)))))

CROSSREFS

Inverse: A249745.

Row 2 of A251722.

Cf. A003961, A007310, A048673, A084967, A126760, A249735, A064216, A249824.

Cf. also A273664, A273669.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 23 2014

STATUS

approved

A347234

Dirichlet convolution of A126760 with A342001.

+20
13

0, 1, 1, 3, 1, 7, 1, 6, 3, 10, 1, 17, 1, 13, 11, 10, 1, 16, 1, 26, 14, 18, 1, 31, 4, 21, 6, 35, 1, 61, 1, 15, 19, 26, 17, 36, 1, 29, 22, 49, 1, 82, 1, 50, 28, 34, 1, 49, 5, 36, 27, 59, 1, 28, 22, 67, 30, 42, 1, 139, 1, 45, 37, 21, 25, 117, 1, 74, 35, 127, 1, 63, 1, 53, 40, 83, 25, 138, 1, 79, 10, 58, 1, 190, 30, 61

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = Sum_{d|n} A126760(d) * A342001(n/d).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A003557(n) = (n/factorback(factorint(n)[, 1]));

A342001(n) = (A003415(n) / A003557(n));

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

A347234(n) = sumdiv(n, d, A126760(d)*A342001(n/d));

CROSSREFS

Cf. A003415, A003557, A126760, A342001.

Cf. also A346485, A347233, A347235, A347395.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 26 2021

STATUS

approved

A347233

Möbius transform of A126760.

+20
12

1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 7, 0, 0, 0, 9, 0, 10, 0, 0, 0, 8, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 14, 0, 0, 0, 17, 0, 14, 0, 0, 0, 19, 0, 20, 0, 0, 0, 16, 0, 22, 0, 0, 0, 23, 0, 24, 0, 0, 0, 20, 0, 26, 0, 0, 0, 27, 0, 22, 0, 0, 0, 29, 0, 24, 0, 0, 0, 24, 0, 32

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

LINKS

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

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = Sum_{d|n} A008683(n/d) * A126760(d).

MATHEMATICA

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

PROG

(PARI)

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

A347233(n) = sumdiv(n, d, moebius(n/d)*A126760(d));

CROSSREFS

Cf. A008683, A126760.

Cf. A000004, A349339 (even and odd bisection).

Cf. also A323881, A346485, A347234, A349136, A349391, A349392, A349393, A349395.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 26 2021

STATUS

approved

A349390

Dirichlet convolution of A126760 with Kimberling's paraphrases, A003602.

+20
12

1, 2, 3, 3, 5, 6, 7, 4, 8, 10, 10, 9, 12, 14, 17, 5, 15, 16, 17, 15, 24, 20, 20, 12, 28, 24, 22, 21, 25, 34, 27, 6, 35, 30, 47, 24, 32, 34, 42, 20, 35, 48, 37, 30, 50, 40, 40, 15, 54, 56, 53, 36, 45, 44, 71, 28, 60, 50, 50, 51, 52, 54, 71, 7, 84, 70, 57, 45, 71, 94, 60, 32, 62, 64, 100, 51, 99, 84, 67, 25, 63, 70, 70

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

FORMULA

a(n) = Sum_{d|n} A126760(n/d) * A003602(d).

MATHEMATICA

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, f[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

PROG

(PARI)

A003602(n) = (1+(n>>valuation(n, 2)))/2;

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

A349390(n) = sumdiv(n, d, A126760(n/d)*A003602(d));

CROSSREFS

Cf. A003602, A126760.

Cf. A347233, A347234, A349391, A349392, A349393, A349395, A349431, A349444, A349447 for other Dirichlet convolutions of A126760. And also A349370.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 15 2021

STATUS

approved

A349393

Inverse Möbius transform of A126760.

+20
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

FORMULA

a(n) = Sum_{d|n} A126760(d).

MATHEMATICA

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

PROG

(PARI)

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

A349393(n) = sumdiv(n, d, A126760(d));

(PARI) a(n)=my(a=valuation(n, 2), b=valuation(n, 3), c=(a+1)*(b+1)); sumdiv(n/3^b>>a, d, d\6*2+d%3)*c; \\ Charles R Greathouse IV, Nov 16 2021

CROSSREFS

Cf. A347233, A347234, A349390, A349391, A349392, A349395 for other Dirichlet convolutions of A126760. And also A349371.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 16 2021

STATUS

approved

A323882

Sum of A126760 and its Dirichlet inverse.

+20
9

2, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, 1, 0, 6, 4, 1, 0, 1, 0, 2, 6, 8, 0, 1, 4, 10, 1, 3, 0, 0, 0, 1, 8, 12, 12, 1, 0, 14, 10, 2, 0, 0, 0, 4, 2, 16, 0, 1, 9, 14, 12, 5, 0, 1, 16, 3, 14, 20, 0, 2, 0, 22, 3, 1, 20, 0, 0, 6, 16, 12, 0, 1, 0, 26, 14, 7, 24, 0, 0, 2, 1, 28, 0, 3, 24, 30, 20, 4, 0, 2, 30, 8, 22, 32, 28, 1, 0, 25, 4, 9, 0, 0, 0, 5, 12

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

From Antti Karttunen, Aug 18 2021: (Start)

No negative terms in range 1 .. 2^20.

Apparently zeros occur only on (some of the) positions given by A030059, with exceptions for example on n = 70, 105, 110, 130, 154, etc, where a(n) > 0.

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = A126760(n) + A323881(n).

For n > 1, a(n) = -Sum_{d|n, 1<d<n} A126760(d) * A323881(n/d). - Antti Karttunen, Aug 18 2021

PROG

(PARI)

up_to = 20000;

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v

v323881 = DirInverseCorrect(vector(up_to, n, A126760(n)));

A323881(n) = v323881[n];

A323882(n) = (A126760(n)+A323881(n));

CROSSREFS

Cf. A030059, A126760, A323881, A323884, A323885, A323887.

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Feb 08 2019

STATUS

approved

A323881

Dirichlet inverse of A126760.

+20
8

1, -1, -1, 0, -2, 1, -3, 0, 0, 2, -4, 0, -5, 3, 2, 0, -6, 0, -7, 0, 3, 4, -8, 0, -5, 5, 0, 0, -10, -2, -11, 0, 4, 6, 0, 0, -13, 7, 5, 0, -14, -3, -15, 0, 0, 8, -16, 0, -8, 5, 6, 0, -18, 0, -3, 0, 7, 10, -20, 0, -21, 11, 0, 0, -2, -4, -23, 0, 8, 0, -24, 0, -25, 13, 5, 0, -2, -5, -27, 0, 0, 14, -28, 0, -5, 15, 10, 0, -30, 0, -1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

MATHEMATICA

b[n_] := b[n] = Which[n == 0, 0, 0 < n < 4, 1, EvenQ[n], b[n/2], Mod[n, 3] == 0, b[n/3], Mod[n, 6] == 1, (n-1)/3 + 1, Mod[n, 6] == 5, (n-5)/3 + 2];

a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];

Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)

PROG

(PARI)

up_to = 20000;

DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v (correctly!)

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

v323881 = DirInverseCorrect(vector(up_to, n, A126760(n)));

A323881(n) = v323881[n];

CROSSREFS

Cf. A126760, A323882, A323883, A323886.

KEYWORD

sign

AUTHOR

Antti Karttunen, Feb 08 2019

STATUS

approved

A349395

Dirichlet convolution of A126760 with Liouville's lambda.

+20
8

1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 4, 0, 0, 1, 5, 0, 6, 1, 0, 0, 7, 0, 8, 0, 0, 2, 9, 0, 10, 0, 0, 0, 8, 1, 12, 0, 0, 0, 13, 0, 14, 3, 1, 0, 15, 0, 15, 0, 0, 4, 17, 0, 14, 0, 0, 0, 19, 0, 20, 0, 2, 1, 16, 0, 22, 5, 0, 0, 23, 0, 24, 0, 0, 6, 20, 0, 26, 1, 1, 0, 27, 0, 22, 0, 0, 0, 29, 0, 24, 7, 0, 0, 24, 0, 32, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

MATHEMATICA

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * LiouvilleLambda[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

PROG

(PARI)

A008836(n) = ((-1)^bigomega(n));

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

A349395(n) = sumdiv(n, d, A126760(n/d)*A008836(d));

CROSSREFS

Cf. A008836, A126760.

Cf. A347233, A347234, A349390, A349391, A349392, A349393 for other Dirichlet convolutions of A126760. And also A349375.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 15 2021

STATUS

approved

A254102

Square array A(row,col) = A253887(A254055(row,col)) = A126760(A254101(row,col)).

+20
7

1, 1, 1, 1, 1, 2, 1, 4, 8, 3, 3, 6, 1, 6, 14, 1, 2, 9, 32, 68, 21, 2, 5, 20, 50, 24, 7, 122, 1, 10, 26, 4, 75, 284, 608, 183, 5, 12, 15, 39, 176, 446, 107, 456, 1094, 2, 7, 5, 86, 230, 132, 669, 2552, 5468, 1641, 1, 4, 38, 104, 129, 345, 1580, 4010, 1914, 2051, 9842

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

Starting with an odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 is found in A254051(row+1,col), and after iterated [i.e., we divide all powers of 2 out] Collatz step: x_new <- A139391(x) = A000265(3x+1) the resulting odd number x_new is located A135764(1,A254055(row+1,col)).

What the resulting odd number will be, is given by A254101(row+1,col) = A000265(A254051(row+1,col)).

That number's column index in array A135765 is then given by A(row+1,col).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Index entries for sequences related to 3x+1 (or Collatz) problem

FORMULA

A(row,col) = A126760(A254051(row,col)) = A126760(A254101(row,col)).

A(row,col) = A253887(A254055(row,col)).

A(row+1,col) = A254048(A135765(row,col)).

EXAMPLE

The top left corner of the array:

1, 1, 1, 1, 3, 1, 2, 1, 5, 2, 1,

1, 1, 4, 6, 2, 5, 10, 12, 7, 4, 16,

2, 8, 1, 9, 20, 26, 15, 5, 38, 44, 12,

3, 6, 32, 50, 4, 39, 86, 104, 57, 17, 140,

14, 68, 24, 75, 176, 230, 129, 78, 338, 392, 53,

21, 7, 284, 446, 132, 345, 770, 932, 507, 294, 1256,

122, 608, 107, 669, 1580, 2066, 1155, 44, 3038, 3524, 942,

183, 456, 2552, 4010, 593, 3099, 6926, 8384, 4557, 331, 11300,

1094, 5468, 1914, 6015, 14216, 18590, 10389, 6288, 27338, 31712, 530,

etc.

PROG

(Scheme)

(define (A254102 n) (A254102bi (A002260 n) (A004736 n)))

;; In turn using either one of these three bivariate functions:

(define (A254102 n) (A254102bi (A002260 n) (A004736 n)))

(define (A254102bi row col) (A126760 (A254051bi row col)))

(define (A254102bi row col) (A253887 (A254055bi row col)))

(define (A254102bi row col) (A126760 (A254101bi row col)))

CROSSREFS

Cf. A000265, A126760, A253887, A254048.

Related arrays: A135764, A135765, A254051, A254055, A254101.

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Jan 28 2015

STATUS

approved

A349392

Dirichlet convolution of A126760 with tau (number of divisors function).

+20
7

1, 3, 3, 6, 4, 9, 5, 10, 6, 12, 6, 18, 7, 15, 12, 15, 8, 18, 9, 24, 15, 18, 10, 30, 16, 21, 10, 30, 12, 36, 13, 21, 18, 24, 26, 36, 15, 27, 21, 40, 16, 45, 17, 36, 24, 30, 18, 45, 26, 48, 24, 42, 20, 30, 35, 50, 27, 36, 22, 72, 23, 39, 30, 28, 40, 54, 25, 48, 30, 78, 26, 60, 27, 45, 48, 54, 44, 63, 29, 60, 15, 48

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

FORMULA

a(n) = Sum_{d|n} A126760(n/d) * A000005(d).

MATHEMATICA

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

PROG

(PARI)

A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

A349392(n) = sumdiv(n, d, A126760(n/d)*numdiv(d));

CROSSREFS

Cf. A000005, A126760.

Cf. A347233, A347234, A349390, A349391, A349393, A349395 for other Dirichlet convolutions of A126760. And also A349372.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 15 2021

STATUS

approved