A113415 -id:A113415 - cp's OEIS frontend

A349915 Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

1, -1, -3, 0, -4, 3, -5, 0, 1, 4, -7, 0, -8, 5, 10, 0, -10, -1, -11, 0, 12, 7, -13, 0, -1, 8, -1, 0, -16, -10, -17, 0, 16, 10, 14, 0, -20, 11, 18, 0, -22, -12, -23, 0, -2, 13, -25, 0, -5, 1, 22, 0, -28, 1, 18, 0, 24, 16, -31, 0, -32, 17, -2, 0, 20, -16, -35, 0, 28, -14, -37, 0, -38, 20, 5, 0, 20, -18, -41, 0, -2, 22

Offset: 1

Antti Karttunen, Dec 07 2021

sign

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

Cf. A000593, A001227, A113415, A349916.

s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < 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)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A113415(n/d) * a(d).

a(n) = A349916(n) - A113415(n).

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).

A366873 a(n) = A113415(A163511(n)), where A113415(n) is the average of number of and sum of odd divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 8, 3, 4, 1, 22, 8, 17, 3, 14, 4, 5, 1, 63, 22, 80, 8, 65, 17, 30, 3, 42, 14, 26, 4, 18, 5, 7, 1, 185, 63, 393, 22, 316, 80, 202, 8, 206, 65, 174, 17, 117, 30, 68, 3, 124, 42, 127, 14, 100, 26, 50, 4, 55, 18, 38, 5, 26, 7, 8, 1, 550, 185, 1956, 63, 1567, 393, 1403, 22, 1020, 316, 1204, 80, 804, 202

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

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

Cf. A113415, A163511, A366874 (rgs-transform).

Cf. also A324186, A366797, A366875.

A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
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));
A366873(n) = A113415(A163511(n));

a(n) = (1/2) * (A324186(n)+A366797(n)).

A353417 a(n) = A113415(A252463(n)), where A113415 is the arithmetic mean between the number of odd divisors and their sum, and A252463 is the hybrid shift.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 4, 1, 1, 4, 5, 3, 7, 5, 3, 1, 8, 8, 10, 4, 4, 7, 11, 3, 8, 8, 1, 5, 13, 14, 16, 1, 5, 10, 14, 8, 17, 11, 7, 4, 20, 18, 22, 7, 3, 13, 23, 3, 17, 17, 8, 8, 25, 22, 18, 5, 10, 16, 28, 14, 31, 17, 4, 1, 26, 26, 32, 10, 11, 26, 35, 8, 37, 20, 8, 11, 26, 30, 38, 4, 1, 22, 41, 18, 30, 23, 13, 7, 43, 42

Offset: 1

Antti Karttunen, Apr 20 2022

nonn

Arithmetic mean between A320107(n) and A353416(n).

Cf. A000203, A064989, A113415 (even bisection), A252463, A320107, A353412, A353416.

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A252463(n) = if(!(n%2),n/2,A064989(n));
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
A353417(n) = A113415(A252463(n));

a(n) = A113415(A252463(n)) = A113415(A353412(n)).
a(n) = (1/2) * (A320107(n) + A353416(n)).
a(2*n) = A113415(n).

A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).

The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

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));

A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

Original entry on oeis.org

1, 2, 5, 4, 8, 10, 11, 8, 20, 16, 17, 20, 20, 22, 42, 16, 26, 40, 29, 32, 58, 34, 35, 40, 53, 40, 74, 44, 44, 84, 47, 32, 90, 52, 94, 80, 56, 58, 106, 64, 62, 116, 65, 68, 174, 70, 71, 80, 102, 106, 138, 80, 80, 148, 146, 88, 154, 88, 89, 168, 92, 94, 241, 64, 172

Offset: 1

Ilya Gutkovskiy, Oct 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A328203, Sequence Machine.

Cf. A000005, A000079 (fixed points), A000203, A001227, A002131, A003602, A006519, A006918, A026741, A038040, A109168, A113415, A140472, A152211, A193356, A245579, A349353 (Dirichlet inverse), A349354.

Cf. also A347957.

a:=[]; for k in [1..65] do if IsOdd(k) then a[k]:=(k * #Divisors(k) + DivisorSigma(1,k)) / 2; else a[k]:=(k * (#Divisors(k) - #Divisors(k div 2)) + DivisorSigma(1,k) - DivisorSigma(1,k div 2)) / 2;  end if; end for; a; // Marius A. Burtea, Oct 07 2019

nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]

A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2)); \\ Antti Karttunen, Nov 13 2021

a(n) = (n * d(n) + sigma(n)) / 2 if n odd, (n * (d(n) - d(n/2)) + sigma(n) - sigma(n/2)) / 2 if n even.
a(n) = (n * A001227(n) + A002131(n)) / 2.
a(2*n) = 2 * a(n).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:
a(n) = Sum_{d|n} A003602(d) * A026741(n/d).
a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.
(End)

A363901 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^2.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 3, 1, 5, 1, 3, 6, 4, 1, 9, 1, 1, 8, 7, 4, 9, 1, 3, 10, 6, 1, 16, 1, 5, 12, 9, 1, 13, 4, 3, 14, 8, 6, 21, 1, 4, 16, 11, 1, 17, 1, 9, 21, 14, 1, 26, 1, 1, 20, 16, 8, 21, 1, 7, 22, 12, 4, 31, 6, 9, 24, 15, 1, 32, 1, 3, 26, 14, 10, 36, 4, 6, 28, 27, 1, 29, 1, 16, 30, 16, 1, 41, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001817, A078181.

Cf. A113415, A363903.

a[n_] := DivisorSum[n, # + 2 &, Mod[#, 3] == 1 &]/3; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%3==1)*(d+2))/3;
```

a(n) = (1/3) * Sum_{d|n, d==1 mod 3} (d+2) = (2 * A001817(n) + A078181(n))/3.

G.f.: Sum_{k>0} k * x^(3*k-2) / (1 - x^(3*k-2)).

A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of the ordered triplet [A336158(n), A206787(n), A336651(n)].
For all i, j >= 1:
  A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A336390(i) = A336390(j) => A336391(i) = A336391(j),
  a(i) = a(j) => A347374(i) = A347374(j),
  a(i) = a(j) => A351036(i) = A351036(j) => A113415(i) = A113415(j),
  a(i) = a(j) => A351461(i) = A351461(j).
From Antti Karttunen, Nov 23 2023: (Start)
Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j), A336158(i) = A336158(j) and A336467(i) = A336467(j), for all i, j >= 1 (this was the original definition). In any case it holds that a(i) = a(j) => b(i) = b(j) for all i, j >= 1. See comment in A351461.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A000265, A000593, A336158, A336467.
Cf. also A003602, A336390, A336391, A351037, A366891.
Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.
Differs from A351036 for the first time at n=637, where a(637) = 272, while A351036(637) = 261.

up_to = 65539;
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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A336158(n) = A046523(A000265(n));
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d));
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351040(n) = [A336158(n), A206787(n), A336651(n)];
v351040 = rgs_transform(vector(up_to, n, Aux351040(n)));
A351040(n) = v351040[n];

Original definition moved to the comment section and replaced with a definition that is at least as encompassing, and conjectured to be equal to the original one. - Antti Karttunen, Nov 23 2023

A366874 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366873(i) = A366873(j) for all i, j >= 0, where A366873 is the average of number of and sum of odd divisors of n as permuted by A163511.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Restricted growth sequence transform of A366873.

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

Cf. A113415, A163511, A366873.

Cf. also A366798, A366806, A366878.

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; };
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
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));
A366873(n) = A113415(A163511(n));
v366874 = rgs_transform(vector(1+up_to,n,A366873(n-1)));
A366874(n) = v366874[1+n];

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 30, 47, 76, 118, 181, 277, 417, 624, 929, 1367, 2001, 2913, 4210, 6056, 8665, 12328, 17466, 24640, 34600, 48395, 67442, 93625, 129520, 178588, 245429, 336252, 459324, 625613, 849762, 1151150, 1555378, 2096332, 2818630, 3780903, 5060240, 6757633, 9005106, 11975265

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A001935, A001936, A001937, A026007, A035528, A093160, A113415, A156616, A284474, A292037, A295832.

nmax = 44; CoefficientList[Series[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[x^k (1 - (-1)^k x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.
G.f.: exp(Sum_{k>=1} x^k*(1 - (-1)^k*x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3*(7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (12 * (7*Zeta(3))^(1/3)) - Pi^4 / (3024*Zeta(3)) - 1/24) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(11/8) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

cp's OEIS Frontend

A349915 Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

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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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A366873 a(n) = A113415(A163511(n)), where A113415(n) is the average of number of and sum of odd divisors of n.

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A353417 a(n) = A113415(A252463(n)), where A113415 is the arithmetic mean between the number of odd divisors and their sum, and A252463 is the hybrid shift.

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A336840 Inverse Möbius transform of A048673.

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A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

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A363901 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^2.

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A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.