A113415 -id:A113415 - cp's OEIS frontend

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

1, 1, 1, 3, 5, 8, 12, 20, 33, 50, 74, 114, 175, 257, 375, 555, 814, 1171, 1677, 2406, 3435, 4855, 6825, 9591, 13428, 18667, 25851, 35745, 49250, 67544, 92340, 125966, 171345, 232257, 313945, 423470, 569778, 764465, 1023231, 1366827, 1821756, 2422394, 3214318, 4257088, 5627086, 7422941

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A000219, A006950, A113415, A156616, A224364, A263140, A273225, A273226, A274621, A295831.

nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k ((-1)^(k + 1) + 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))/(1 - x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} x^k*((-1)^(k+1) + 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) / (24 * (7*Zeta(3))^(1/3)) - Pi^4 / (12096 * Zeta(3)) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(23/24) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(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 30 2022

Restricted growth sequence transform of the ordered pair [A000593(n), A336158(n)], where A000593(n) is the sum of odd divisors of n, and A336158(n) is the least representative of the prime signature of the odd part of n.
For all i, j:
  A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
  a(i) = a(j) => A113415(i) = A113415(j).

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

Cf. A000265, A000593, A003602, A046523, A113415, A336158.
Cf. also A351037.
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 A351040 for the first time at n=637, where a(637) = 261, while A351040(637) = 272.

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; };
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]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A000593(n) = sigma(A000265(n));
Aux351036(n) = [A000593(n), A336158(n)];
v351036 = rgs_transform(vector(up_to, n, Aux351036(n)));
A351036(n) = v351036[n];

A366798 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366797(i) = A366797(j) for all i, j >= 0, where A366797 is the number of odd divisors permuted by A163511.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 4, 2, 2, 1, 5, 4, 4, 3, 6, 3, 3, 2, 6, 4, 4, 2, 4, 2, 2, 1, 6, 5, 5, 4, 7, 4, 4, 3, 8, 6, 6, 3, 6, 3, 3, 2, 7, 6, 6, 4, 7, 4, 4, 2, 6, 4, 4, 2, 4, 2, 2, 1, 9, 6, 6, 5, 10, 5, 5, 4, 11, 7, 7, 4, 7, 4, 4, 3, 11, 8, 8, 6, 11, 6, 6, 3, 8, 6, 6, 3, 6, 3, 3, 2, 10, 7, 7, 6, 11, 6, 6

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Restricted growth sequence transform of A366797.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A113415, A163511, A366797.

Cf. also A366806, 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; };
A001227(n) = numdiv(n>>valuation(n, 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));
A366797(n) = A001227(A163511(n));
v366798 = rgs_transform(vector(1+up_to,n,A366797(n-1)));
A366798(n) = v366798[1+n];

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.

Original entry on oeis.org

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));

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];

A363902 Expansion of Sum_{k>0} x^(2k) / (1 - x^(3k))^2.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 4, 0, 3, 4, 1, 0, 6, 2, 4, 6, 1, 0, 10, 0, 5, 8, 4, 2, 10, 0, 6, 10, 3, 0, 15, 4, 7, 14, 1, 0, 14, 0, 13, 14, 6, 0, 20, 2, 9, 16, 4, 0, 20, 6, 10, 18, 1, 6, 28, 0, 11, 20, 10, 0, 22, 0, 15, 24, 5, 0, 30, 8, 20, 24, 4, 0, 26, 2, 14, 30, 10, 0, 40, 0, 15, 28, 6, 8

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A001822, A078182, A363901.

Cf. A113415, A363904.

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

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

a(n) = (1/3) * Sum_{d|n, d==2 mod 3} (d+1) = (A001822(n) + A078182(n))/3.

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

A363904 Expansion of Sum_{k>0} x^(3k) / (1 - x^(4k))^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 0, 2, 5, 0, 0, 1, 5, 0, 3, 3, 6, 1, 0, 0, 8, 2, 0, 5, 8, 0, 4, 0, 11, 1, 0, 5, 11, 0, 0, 3, 11, 3, 5, 6, 12, 1, 2, 0, 14, 0, 0, 8, 17, 2, 6, 0, 15, 5, 0, 8, 19, 0, 0, 4, 17, 0, 7, 11, 18, 1, 0, 0, 24, 5, 5, 11, 20, 0, 8, 0, 21, 3, 0, 11, 23, 3, 0, 5, 25, 6

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A001842, A050452, A363903.

Cf. A113415, A363902.

a[n_] := DivisorSum[n, # + 1 &, Mod[#, 4] == 3 &]/4; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%4==3)*(d+1))/4;
```

a(n) = (1/4) * Sum_{d|n, d==3 mod 4} (d+1) = (A001842(n) + A050452(n))/4.

G.f.: Sum_{k>0} k * x^(4*k-1) / (1 - x^(4*k-1)).

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];

A363969 Expansion of Sum_{k>0} k^2 * x^(2k-1) / (1 - x^(2k-1)).

Original entry on oeis.org

1, 1, 5, 1, 10, 5, 17, 1, 30, 10, 37, 5, 50, 17, 78, 1, 82, 30, 101, 10, 142, 37, 145, 5, 179, 50, 226, 17, 226, 78, 257, 1, 330, 82, 350, 30, 362, 101, 454, 10, 442, 142, 485, 37, 632, 145, 577, 5, 642, 179, 762, 50, 730, 226, 830, 17, 946, 226, 901, 78, 962, 257, 1191, 1, 1148, 330, 1157, 82

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001227, A113415.

Cf. A050999, A363974.

a[n_] := DivisorSum[n, ((# + 1)/2)^2 &, OddQ[#] &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%2==1)*((d+1)/2)^2);

a(n) = Sum_{d|n, d==1 mod 2} ((d+1)/2)^2.

A363974 Expansion of Sum_{k>0} x^k / (1 - x^(2*k))^3.

Original entry on oeis.org

1, 1, 4, 1, 7, 4, 11, 1, 19, 7, 22, 4, 29, 11, 46, 1, 46, 19, 56, 7, 80, 22, 79, 4, 98, 29, 124, 11, 121, 46, 137, 1, 178, 46, 188, 19, 191, 56, 242, 7, 232, 80, 254, 22, 337, 79, 301, 4, 336, 98, 400, 29, 379, 124, 434, 11, 494, 121, 466, 46, 497, 137, 623, 1, 596, 178, 596, 46, 712, 188, 667, 19

Offset: 1

Seiichi Manyama, Jun 30 2023

nonn

Cf. A001227, A113415.

Cf. A363969.

a[n_] := DivisorSum[n, Binomial[(#+1)/2+1,2] &, OddQ[#] &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%2==1)*binomial((d+1)/2+1, 2));

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

a(n) = Sum_{d|n, d==1 mod 2} binomial((d+1)/2+1,2).

cp's OEIS Frontend

A295832 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.

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A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A366798 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366797(i) = A366797(j) for all i, j >= 0, where A366797 is the number of odd divisors permuted by A163511.

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

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A363902 Expansion of Sum_{k>0} x^(2*k) / (1 - x^(3*k))^2.

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A363904 Expansion of Sum_{k>0} x^(3*k) / (1 - x^(4*k))^2.

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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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A363969 Expansion of Sum_{k>0} k^2 * x^(2*k-1) / (1 - x^(2*k-1)).

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A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

A363902 Expansion of Sum_{k>0} x^(2k) / (1 - x^(3k))^2.

A363904 Expansion of Sum_{k>0} x^(3k) / (1 - x^(4k))^2.

A363969 Expansion of Sum_{k>0} k^2 * x^(2k-1) / (1 - x^(2k-1)).