A293549 -id:A293549 - cp's OEIS frontend

A293548 Expansion of Product_{k>=2} 1/(1 - x^k)^omega(k), where omega(k) is the number of distinct primes dividing k (A001221).

1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 15, 16, 28, 29, 46, 54, 77, 90, 131, 150, 211, 251, 337, 401, 540, 637, 839, 1006, 1296, 1551, 1995, 2373, 3013, 3610, 4523, 5410, 6754, 8045, 9965, 11897, 14614, 17410, 21313, 25316, 30816, 36615, 44307, 52539, 63387, 74975, 90078

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Euler transform of A001221.

N. J. A. Sloane, Transforms

Cf. A001221, A006171, A293549.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeNu[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

G.f.: Product_{k>=2} 1/(1 - x^k)^b(k), where b(k) = [x^k] Sum_{j>=1} x^prime(j)/(1 - x^prime(j)).

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*omega(d).

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, -3, 4, -4, 4, -9, 14, -19, 30, -42, 50, -76, 128, -194, 286, -412, 598, -909, 1386, -2100, 3178, -4763, 7122, -10758, 16414, -25061, 38056, -57643, 87568, -133436, 203618, -311128, 475536, -726355, 1109718, -1697766, 2601166, -3987903, 6114666

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):
seq(a(n), n = 0..43); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[EulerPhi,30]]

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, -1, -1, 0, 1, 0, -1, -1, -1, 1, 3, 3, -2, -5, -4, 0, 7, 7, 0, -9, -10, 2, 15, 15, -3, -27, -30, 3, 46, 51, 1, -71, -91, -7, 117, 157, 23, -194, -265, -57, 318, 465, 111, -536, -821, -230, 893, 1456, 505, -1485, -2559, -1036, 2433, 4483, 2022

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320777, A320778, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-NumberOfPrimeFactors(n))):
seq(a(n), n = 0..59); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeOmega,100]]

A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, -1, -2, 1, 3, 1, -2, -2, 1, 0, -4, 0, 6, 6, -4, -8, 1, 4, -4, -5, 10, 16, -4, -25, -7, 17, 5, -16, 2, 42, 12, -58, -48, 40, 59, -27, -44, 67, 86, -103, -187, 36, 236, 45, -213, -5, 284, -23, -526, -188, 663, 520

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeNu,100]]

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

Original entry on oeis.org

1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -2, 3, 0, -1, -3, 6, -3, 0, -6, 12, -6, 0, -9, 23, -17, 0, -15, 47, -40, 8, -24, 91, -101, 34, -46, 181, -230, 109, -92, 354, -534, 323, -208, 690, -1177, 883, -520, 1365, -2603, 2297, -1377, 2760, -5641, 5789, -3721, 5741

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Table[Abs[MoebiusMu[n]],{n,30}]]

A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 10, 23, 32, 64, 98, 187, 296, 543, 891, 1595, 2660, 4694, 7924, 13854, 23556, 40940, 69939, 121122, 207490, 358517, 615292, 1061635, 1824013, 3144404, 5406257, 9314645, 16021922, 27595176, 47478950, 81757104, 140691461, 242232918, 416890645, 717712748, 1235289624

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001222.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001222, A022559, A112967, A129921, A180305, A293549, A300671.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[bigomega](i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 41; CoefficientList[Series[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/(1 - Sum[PrimeOmega[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeOmega[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 41}]

G.f.: 1/(1 - Sum_{k>=2} A001222(k)*x^k).

A320784 Negated inverse Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 3, 3, 5, 8, 11, 14, 23, 31, 47, 68, 101, 144, 217, 315, 471, 693, 1035, 1528, 2287, 3397, 5085, 7587, 11377, 17017, 25565, 38349, 57681, 86724, 130645, 196778, 296853, 447864, 676479, 1022082, 1545685, 2338299, 3540111, 5361606, 8125551

Offset: 0

Gus Wiseman, Oct 22 2018

nonn

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
-EulerInvTransform[-Table[SquaresR[1,8*n+1]/2,{n,30}]]

A320783 Inverse Euler transform of (-1)^(n - 1).

Original entry on oeis.org

1, 1, -2, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986, -505294125, 981706806

Offset: 0

Gus Wiseman, Oct 22 2018

sign

After a(1) and a(2), same as A038063.

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[(-1)^(#-1)&,30]]

A320785 Inverse Euler transform of the number of factorizations function A001055.

Original entry on oeis.org

1, 1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 1, -1, 0, 0, 1, 1, -3, 3, -3, 0, 4, -6, 6, -5, 5, -1, -7, 13, -16, 15, -8, -3, 12, -25, 41, -40, 21, 10, -51, 83, -93, 81, -38, -44, 148, -234, 258, -190, 35, 184, -429, 616, -660, 480, -18, -640, 1289, -1714, 1693, -1039, -268

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
EulerInvTransform[Table[Length[facs[n]],{n,100}]]

A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -2, 2, -2, 3, -5, 6, -7, 11, -16, 20, -27, 39, -55, 75, -102, 145, -207, 286, -397, 565, -802, 1123, -1581, 2248, -3193, 4517, -6399, 9112, -12984, 18457, -26270, 37502, -53553, 76416, -109146, 156135, -223446, 319764, -457884, 656288, -941081

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[PadRight[{1,0,1},50]]

cp's OEIS Frontend

A293548 Expansion of Product_{k>=2} 1/(1 - x^k)^omega(k), where omega(k) is the number of distinct primes dividing k (A001221).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320784 Negated inverse Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A320783 Inverse Euler transform of (-1)^(n - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A320785 Inverse Euler transform of the number of factorizations function A001055.

Views

Author

Keywords

Comments