A293548 -id:A293548 - cp's OEIS frontend

A293549 Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

1, 0, 1, 1, 3, 2, 6, 5, 13, 12, 23, 24, 47, 47, 82, 92, 152, 167, 265, 301, 462, 532, 779, 914, 1324, 1548, 2174, 2590, 3573, 4250, 5771, 6904, 9254, 11092, 14638, 17606, 23043, 27680, 35820, 43155, 55383, 66642, 84850, 102141, 129171, 155394, 195134, 234679, 293184, 352096, 437359

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Euler transform of A001222.
Comment from R. J. Mathar, Sep 10 2021 (Start):
The triangle of the multiset transformation of A001222 looks as follows:
;1
0 ;0
1 0 ;1
1 0 0 ;1
2 1 0 0 ;3
1 1 0 0 0 ;2
2 3 1 0 0 0 ;6
1 3 1 0 0 0 0 ;5
3 6 3 1 0 0 0 0 ;13
2 5 4 1 0 0 0 0 0 ;12
2 9 8 3 1 0 0 0 0 0 ;23
1 9 9 4 1 0 0 0 0 0 0 ;24
3 14 17 9 3 1 0 0 0 0 0 0 ;47
1 12 18 11 4 1 0 0 0 0 0 0 0 ;47
2 17 29 21 9 3 1 0 0 0 0 0 0 0 ;82
...
The second column is A001222, the row sums (after the semicolons) are this sequence. (End)

N. J. A. Sloane, Transforms

Cf. A001222, A006171, A293548.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeOmega[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeOmega[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_{p prime, j>=1} x^(p^j)/(1 - x^(p^j)).

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

A328260 a(n) = n * omega(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 24, 13, 28, 30, 16, 17, 36, 19, 40, 42, 44, 23, 48, 25, 52, 27, 56, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 80, 41, 126, 43, 88, 90, 92, 47, 96, 49, 100, 102, 104, 53, 108, 110, 112, 114, 116, 59, 180, 61, 124, 126, 64, 130, 198, 67, 136, 138, 210

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A001222, A007947, A038040, A061397, A066959, A069359, A246655 (fixed points), A293548, A298473.

[0] cat [n*(#PrimeDivisors(n)):n in [2..70]]; // Marius A. Burtea, Oct 10 2019

Table[n PrimeNu[n], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n)=n*omega(n) \\ Charles R Greathouse IV, Mar 16 2022

G.f.: Sum_{k>=1} prime(k) * x^prime(k) / (1 - x^prime(k))^2.
a(n) = bigomega(rad(n)^n).
a(n) = Sum_{d|n} A061397(n/d) * d.
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/log log x. - Charles R Greathouse IV, Mar 16 2022

A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 8, 15, 23, 40, 63, 108, 172, 290, 471, 782, 1280, 2119, 3474, 5741, 9432, 15557, 25590, 42180, 69413, 114371, 188276, 310136, 510637, 841045, 1384883, 2280831, 3755862, 6185457, 10185941, 16774695, 27624215, 45492412, 74916559, 123374127, 203172520, 334587577

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001221.

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

Cf. A001221, A013939, A112965, A129921, A180305, A293548, A300672.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*nops(ifactors(i)[2]), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 42; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[PrimeNu[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeNu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

G.f.: 1/(1 - Sum_{k>=2} A001221(k)*x^k).

A319130 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^omega(k)), where omega(k) = number of distinct primes dividing k (A001221).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 31, 47, 81, 125, 203, 305, 482, 710, 1082, 1582, 2348, 3380, 4933, 7007, 10048, 14136, 19972, 27796, 38822, 53510, 73903, 101033, 138165, 187351, 254055, 341923, 459956, 614904, 821162, 1090740, 1447109, 1910665, 2519325, 3308019, 4336956

Offset: 0

Ilya Gutkovskiy, Sep 11 2018

nonn

Euler transform of A034444.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A001221, A006171, A008683, A034444, A048250, A293548.

with(numtheory): a:=series(mul(1/(1-x^k)^(2^nops(factorset(k))),k=1..50),x=0,41): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Apr 02 2019

nmax = 40; CoefficientList[Series[Product[1/(1 - x^k)^(2^PrimeNu[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d 2^PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A034444(k).
G.f.: exp(Sum_{k>=1} A048250(k)*x^k/(k*(1 - x^k))).
G.f.: exp(Sum_{k>=1} Sum_{j>=1} mu(j)^2*x^(j*k)/(k*(1 - x^(j*k)))), where mu = Möbius function (A008683).
log(a(n)) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 13 2018

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]]

cp's OEIS Frontend

A293549 Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

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

A328260 a(n) = n * omega(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A319130 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^omega(k)), where omega(k) = number of distinct primes dividing k (A001221).

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.