A349915 -id:A349915 - cp's OEIS frontend

A366875 a(n) = A349915(A163511(n)), where A349915 is the Dirichlet inverse of the arithmetic mean between the number and sum of the odd divisors of n.

1, -1, 0, -3, 0, 1, 3, -4, 0, -1, -1, -1, 0, 10, 4, -5, 0, -2, 1, -8, 0, 5, 1, -5, 0, -2, -10, 14, 0, 12, 5, -7, 0, -4, 2, -24, 0, 32, 8, -27, 0, 1, -5, 26, 0, 18, 5, -19, 0, 6, 2, 11, 0, -20, -14, 20, 0, -2, -12, 18, 0, 16, 7, -8, 0, -8, 4, -72, 0, 112, 24, -108, 0, -8, -32, 162, 0, 108, 27, -125, 0, 3, -1, 35, 0

Offset: 0

Antti Karttunen, Oct 27 2023

sign

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A113415, A163511, A349915, A366878 (rgs-transform).

Cf. also A366873.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
A366875(n) = A349915(A163511(n));

A366383 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349915(i) = A349915(j) for all i, j >= 1, where A349915 is Dirichlet inverse of arithmetic mean between the number of odd divisors and their sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 1, 8, 9, 4, 10, 11, 12, 4, 13, 2, 14, 4, 15, 16, 17, 4, 2, 18, 2, 4, 19, 13, 20, 4, 21, 12, 22, 4, 23, 24, 25, 4, 26, 27, 28, 4, 29, 30, 31, 4, 7, 1, 32, 4, 33, 1, 25, 4, 34, 21, 35, 4, 36, 37, 29, 4, 38, 19, 39, 4, 40, 41, 42, 4, 43, 38, 11, 4, 38, 44, 45, 4, 29, 32, 46, 4, 34, 47, 48, 4, 49

Offset: 1

Antti Karttunen, Oct 12 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A113415, A349915.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
v366383 = rgs_transform(DirInverseCorrect(vector(up_to,n,A113415(n))));
A366383(n) = v366383[n];

A113415 Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 5, 1, 8, 4, 7, 3, 8, 5, 14, 1, 10, 8, 11, 4, 18, 7, 13, 3, 17, 8, 22, 5, 16, 14, 17, 1, 26, 10, 26, 8, 20, 11, 30, 4, 22, 18, 23, 7, 42, 13, 25, 3, 30, 17, 38, 8, 28, 22, 38, 5, 42, 16, 31, 14, 32, 17, 55, 1, 44, 26, 35, 10, 50, 26, 37, 8, 38, 20, 65, 11, 50, 30, 41

Offset: 1

Michael Somos, Oct 29 2005

nonn

Arithmetic mean between the number of odd divisors (A001227) and their sum (A000593). This fact was essentially found by the algorithmic search of Jon Maiga's Sequence Machine, and is easily seen to be correct when compared to the PARI-program given by the original author. - Antti Karttunen, Dec 07 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A113415, Sequence Machine.

Cf. A000265, A000593, A001227, A001511, A003602, A048673, A064989, A069734, A336840, A349371, A349915 (Dirichlet inverse).

Quadrisection of A349916.

Array[DivisorSum[#, If[OddQ[#], (# + 1)/2, 0] &] &, 79] (* Michael De Vlieger, Dec 08 2021 *)

a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)))

G.f.: Sum_{k>0} x^k/(1-x^(2k))^2 = Sum_{k>0} k x^(2k-1)/(1-x^(2k-1)).
a(n) = (1/2) * Sum_{d|n} (d+1)*(d mod 2). - Wesley Ivan Hurt, Nov 25 2021 [From PARI prog]
From Antti Karttunen, Dec 07 2021: (Start)
All these formulas, except the last, were found by the Sequence Machine in some form or another:
a(n) = (1/2) * (A000593(n)+A001227(n)).
a(n) = A069734(A000265(n)). [See either Rutherford's or Luschny's formula in A069734]
a(n) = A349371(n) / A001511(n).
a(n) = A349371(A000265(n)) = A336840(A064989(n)).
a(n) = a(2*n) = a(A000265(n)) = A349916(4*n).
(End)

A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

Original entry on oeis.org

2, 0, 0, 1, 0, 6, 0, 1, 9, 8, 0, 3, 0, 10, 24, 1, 0, 7, 0, 4, 30, 14, 0, 3, 16, 16, 21, 5, 0, 4, 0, 1, 42, 20, 40, 8, 0, 22, 48, 4, 0, 6, 0, 7, 40, 26, 0, 3, 25, 18, 60, 8, 0, 23, 56, 5, 66, 32, 0, 14, 0, 34, 53, 1, 64, 10, 0, 10, 78, 12, 0, 8, 0, 40, 70, 11, 70, 12, 0, 4, 61, 44, 0, 18, 80, 46, 96, 7, 0, 44, 80

Offset: 1

Antti Karttunen, Dec 07 2021

nonn

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

Cf. A113415 (also a quadrisection of this sequence), A349915.

Cf. also A349913, A349914.

s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
A349916(n) = (A113415(n)+A349915(n));

a(n) = A113415(n) + A349915(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A113415(d) * A349915(n/d).
For all n >= 1, a(4*n) = A113415(n).

A366878 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366875(i) = A366875(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 1, 5, 6, 3, 2, 2, 2, 3, 7, 8, 9, 3, 10, 1, 11, 3, 12, 1, 9, 3, 10, 13, 14, 3, 15, 12, 16, 3, 6, 17, 18, 3, 19, 20, 21, 3, 1, 9, 22, 3, 23, 12, 24, 3, 25, 17, 26, 3, 27, 28, 29, 3, 10, 30, 23, 3, 31, 32, 11, 3, 11, 8, 33, 3, 34, 35, 36, 3, 11, 37, 38, 3, 39, 40, 41, 3, 5, 2, 42, 3, 43, 44, 45, 3, 1, 46, 47, 3

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Restricted growth sequence transform of A366875.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A113415, A163511, A349915, A366875.

Cf. also A366383, A366874.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
A366875(n) = A349915(A163511(n));
v366878 = rgs_transform(vector(1+up_to,n,A366875(n-1)));
A366878(n) = v366878[1+n];

A378520 Dirichlet inverse of A336840, where A336840 is the inverse Möbius transform of A048673.

Original entry on oeis.org

1, -3, -4, 1, -5, 10, -7, -1, -1, 12, -8, -2, -10, 16, 14, -2, -11, 5, -13, -2, 18, 18, -16, 6, -5, 22, -8, -2, -17, -20, -20, -4, 20, 24, 20, 1, -22, 28, 24, 8, -23, -20, -25, -2, 11, 34, -28, 14, -19, 18, 26, -2, -31, 32, 22, 12, 30, 36, -32, 4, -35, 42, 17, -8, 26, -20, -37, -2, 36, -14, -38, 3, -41, 46, 26, -2, 26

Offset: 1

Antti Karttunen, Nov 30 2024

sign

Möbius transform of A323893.
Dirichlet inverse of A336840.
Cf. A003961, A008683, A349915.

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));
memoA378520 = Map();
A378520(n) = if(1==n,1,my(v); if(mapisdefined(memoA378520,n,&v), v, v = -sumdiv(n,d,if(dA336840(n/d)*A378520(d),0)); mapput(memoA378520,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA336840(n/d) * a(d).
a(n) = Sum_{d|n} A008683(n/d)*A323893(d).
a(n) = A349915(A003961(n)).

cp's OEIS Frontend

A366875 a(n) = A349915(A163511(n)), where A349915 is the Dirichlet inverse of the arithmetic mean between the number and sum of the odd divisors of n.

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A366383 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349915(i) = A349915(j) for all i, j >= 1, where A349915 is Dirichlet inverse of arithmetic mean between the number of odd divisors and their sum.

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A113415 Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.

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A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

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A366878 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366875(i) = A366875(j) for all i, j >= 0.

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A378520 Dirichlet inverse of A336840, where A336840 is the inverse Möbius transform of A048673.

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