A126760 -id:A126760 - cp's OEIS frontend

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

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

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

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

			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.

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

Inverse: A249745.
Row 2 of A251722.
Cf. A003961, A007310, A048673, A084967, A126760, A249735, A064216, A249824.
Cf. also A273664, A273669.

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 *)

(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)))))

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

A347234 Dirichlet convolution of A126760 with A342001.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A126760, A342001.

Cf. also A346485, A347233, A347235, A347395.

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

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

A347233 Möbius transform of A126760.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008683, A126760.
Cf. A000004, A349339 (even and odd bisection).
Cf. also A323881, A346485, A347234, A349136, A349391, A349392, A349393, A349395.

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 *)

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A347233(n) = sumdiv(n,d,moebius(n/d)*A126760(d));

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

A349390 Dirichlet convolution of A126760 with Kimberling's paraphrases, A003602.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A003602, A126760.

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

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 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349390(n) = sumdiv(n,d,A126760(n/d)*A003602(d));

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

A349393 Inverse Möbius transform of A126760.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 16 2021

nonn

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

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

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 *)

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349393(n) = sumdiv(n,d,A126760(d));

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

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

A323882 Sum of A126760 and its Dirichlet inverse.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 08 2019

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)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

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

up_to = 20000;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA126760(n)));
A323881(n) = v323881[n];
A323882(n) = (A126760(n)+A323881(n));

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

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

A323881 Dirichlet inverse of A126760.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 08 2019

sign

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

Cf. A126760, A323882, A323883, A323886.

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 *)

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(dA126760(n) = {n&&n\=3^valuation(n, 3)<A126760
v323881 = DirInverseCorrect(vector(up_to,n,A126760(n)));
A323881(n) = v323881[n];

A349395 Dirichlet convolution of A126760 with Liouville's lambda.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A008836, A126760.

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

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 *)

A008836(n) = ((-1)^bigomega(n));
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349395(n) = sumdiv(n,d,A126760(n/d)*A008836(d));

A254102 Square array A(row,col) = A253887(A254055(row,col)) = A126760(A254101(row,col)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 28 2015

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).

			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.

Cf. A000265, A126760, A253887, A254048.

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

(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)))

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)).

A349392 Dirichlet convolution of A126760 with tau (number of divisors function).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A000005, A126760.

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

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 *)

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349392(n) = sumdiv(n,d,A126760(n/d)*numdiv(d));

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

cp's OEIS Frontend

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

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A347234 Dirichlet convolution of A126760 with A342001.

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A347233 Möbius transform of A126760.

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A349390 Dirichlet convolution of A126760 with Kimberling's paraphrases, A003602.

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A349393 Inverse Möbius transform of A126760.

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A323882 Sum of A126760 and its Dirichlet inverse.

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A323881 Dirichlet inverse of A126760.

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A349395 Dirichlet convolution of A126760 with Liouville's lambda.

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