A305614 -id:A305614 - cp's OEIS frontend

A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

1, 0, -1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -3, 3, -3, 4, -4, 5, -6, 6, -6, 8, -9, 9, -11, 12, -13, 14, -16, 19, -19, 21, -25, 26, -28, 32, -36, 38, -41, 46, -50, 55, -60, 65, -70, 77, -85, 91, -99, 108, -116, 126, -138, 149, -160, 174, -188, 202, -219, 237, -255, 274, -296

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005, A000040, A000586, A000607, A001221, A008683, A083399, A088705, A280954, A305614.

nn=20;
ser=Product[1/(1+x^p),{p,Select[Range[nn],PrimeQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}] (* Gus Wiseman, Jun 06 2018 *)

a(n) = A184198(n) - A184199(n). - Vaclav Kotesovec, Jan 11 2021

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).

			The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.

q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 12, 13, 17, 21, 25, 32, 39, 46, 58, 68, 83, 99, 121, 141, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

			The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A106404 Number of even semiprimes dividing n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0

Offset: 1

Reinhard Zumkeller, May 02 2005

nonn

Also the number of prime divisors p|n such that n/p is even. - Gus Wiseman, Jun 06 2018

			a(60) = #{4, 6, 10} = #{2*2, 2*3, 2*5} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000607, A001221, A001227, A008683, A083399, A088705, A100484, A205745, A305614.

a106404 n = length [d | d <- takeWhile (<= n) a100484_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 31 2012

Table[Length[Select[Divisors[n],PrimeQ[#]&&EvenQ[n/#]&]],{n,100}] (* Gus Wiseman, Jun 06 2018 *)
Table[Count[Divisors[n],?(EvenQ[#]&&PrimeOmega[#]==2&)],{n,110}] (* _Harvey P. Dale, May 04 2021 *)
a[n_] := If[EvenQ[n], PrimeNu[n/2], 0]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

a(n)=if(n%2,0,omega(n/2)) \\ Charles R Greathouse IV, Jan 30 2012

def A106404(n):
    return add(1-(n/d)%2 for d in divisors(n) if is_prime(d))
print([A106404(n) for n in (1..105)])  # Peter Luschny, Jan 30 2012

a(n) = A086971(n) - A106405(n).
a(A100484(n)) = 1.
a(A005408(n)) = 0.
a(A005843(n)) > 0 for n>1.
a(2n) = omega(n), a(2n+1) = 0, where omega(n) is the number of distinct prime divisors of n, A001221. - Franklin T. Adams-Watters, Jun 09 2006
a(n) = card { d | d*p = n, d even, p prime }. - Peter Luschny, Jan 30 2012
O.g.f.: Sum_{p prime} x^(2p)/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

A205745 a(n) = card { d | d*p = n, d odd, p prime }.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2

Offset: 1

Peter Luschny, Jan 30 2012

nonn

Equivalently, a(n) is the number of prime divisors p|n such that n/p is odd. - Gus Wiseman, Jun 06 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000607, A001221, A008683, A010051, A068050, A077761, A083399, A088705, A106404, A305614.

a205745 n = sum $ map ((`mod` 2) . (n `div`))
   [p | p <- takeWhile (<= n) a000040_list, n `mod` p == 0]
-- Reinhard Zumkeller, Jan 31 2012

a[n_] := Sum[ Boole[ OddQ[d] && PrimeQ[n/d] ], {d, Divisors[n]} ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 27 2013 *)

a(n)=if(n%2,omega(n),n%4/2) \\ Charles R Greathouse IV, Jan 30 2012

def A205745(n):
    return sum((n//d) % 2 for d in divisors(n) if is_prime(d))
[A205745(n) for n in (1..105)]

O.g.f.: Sum_{p prime} x^p/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

Sum_{k=1..n} a(k) = (n/2) * (log(log(n)) + B) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A373458 Expansion of Sum_{p prime} x^p/(1 - p*x^p).

Original entry on oeis.org

0, 1, 1, 2, 1, 7, 1, 8, 9, 21, 1, 59, 1, 71, 106, 128, 1, 499, 1, 637, 778, 1035, 1, 4235, 625, 4109, 6561, 8535, 1, 39192, 1, 32768, 59170, 65553, 18026, 308219, 1, 262163, 531610, 602413, 1, 2659706, 1, 2098483, 5173594, 4194327, 1, 22737515, 117649, 18730341

Offset: 1

Seiichi Manyama, Jun 06 2024

Cf. A001221, A305614, A373459.

A373458 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isprime(d) then
            a := a+d^(n/d-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A373458(n),n=1..20) ; # R. J. Mathar, Jun 07 2024

a[n_]:=Sum[Boole[PrimeQ[d]]d^(n/d-1),{d,Divisors[n]}]; Array[a,50] (* Stefano Spezia, Mar 30 2025 *)

a(n) = sumdiv(n, d, isprime(d)*d^(n/d-1));

a(n) = Sum_{p|n prime} p^(n/p - 1).

If p is prime, a(p) = 1.

A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 2, 4, 3, 2, 4, 6, 5, 4, 7, 10, 8, 7, 11, 15, 13, 12, 17, 22, 19, 18, 25, 30, 28, 26, 35, 42, 39, 38, 49, 59, 56, 54, 69, 81, 77, 76, 94, 110, 105, 105, 127, 147, 141, 142, 171, 195, 189, 190, 227, 257, 250, 254, 299, 335, 328, 334, 390, 432

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

			O.g.f.: 1/((1 - x)(1 + x^2)(1 - x^3)(1 - x^5)(1 + x^6)(1 - x^7)(1 + x^10)...).

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1+(-x)^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305633 Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).

Original entry on oeis.org

0, 0, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -2, 1, 1, 3, -1, 1, 2, 1, -2, 3, 1, 1, -3, 1, 1, 1, -2, 1, 1, 1, -1, 3, 1, 3, -3, 1, 1, 3, -3, 1, 1, 1, -2, 4, 1, 1, -4, 1, 2, 3, -2, 1, 3, 3, -3, 3, 1, 1, -4, 1, 1, 4, -1, 3, 1, 1, -2, 3, 1, 1, -3, 1, 1, 4, -2, 3, 1, 1, -4

Offset: 0

Gus Wiseman, Jun 07 2018

sign

Cf. A000607, A001597, A007916, A048165, A081362, A091050, A303707, A304779, A304817, A305614, A305630-A305635.

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Sum[x^p/(1+x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

A317528 Expansion of Sum_{k>=1} mu(k)^2*x^k/(1 + x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 2, -2, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 4, -2, 2, 0, 2, -4, 4, 0, 2, -4, 2, 0, 2, -4, 2, 0, 2, -2, 4, 0, 4, -4, 2, 0, 4, -4, 2, 0, 2, -4, 4, 0, 2, -4, 2, 0, 4, -4, 2, 0, 4, -4, 4, 0, 2, -8, 2, 0, 4, -2, 4, 0, 2, -4, 4, 0, 2, -4, 2, 0, 4, -4, 4, 0, 2, -4, 2, 0, 2, -8, 4, 0, 4, -4, 2, 0, 4, -4, 4, 0, 4

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005117, A008683, A008966, A034444, A048272, A100007, A300894, A305614.

with(numtheory): seq(coeff(series(add(mobius(k)^2*x^k/(1+x^k),k=1..n), x,n+1),x,n),n=1..120); # Muniru A Asiru, Jul 30 2018

nmax = 95; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, SquareFreeQ[#] &], {n, 95}]
f[p_, e_] := 2; f[2, e_] := If[e == 1, 0, -2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2022 *)

A317528(n) = sumdiv(n,d,((-1)^(1+d))*issquarefree(n/d)); \\ Antti Karttunen, Dec 05 2021

G.f.: Sum_{k>=1} x^A005117(k)/(1 + x^A005117(k)).
L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A008966(d).
If n is odd, a(n) = A034444(n).
Multiplicative with a(2) = 0, a(2^e) = -2 for e>1, and a(p^e) = 2 for p>2 and e>=1. - Amiram Eldar, Nov 19 2022

A317531 Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 2, -2, 1, -1, 1, -1, 2, 0, 1, -2, 2, 0, 3, -1, 1, -1, 1, -3, 2, 0, 2, -2, 1, 0, 2, -2, 1, -1, 1, -1, 3, 0, 1, -3, 2, -1, 2, -1, 1, -2, 2, -2, 2, 0, 1, -2, 1, 0, 3, -4, 2, -1, 1, -1, 2, -1, 1, -3, 1, 0, 3, -1, 2, -1, 1, -3, 4, 0, 1, -2, 2, 0, 2, -2, 1, -2, 2, -1, 2, 0, 2

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

sign

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

Cf. A001222, A048272, A069513, A246655, A288571, A305614, A317528.

nmax = 95; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + Boole[PrimePowerQ[k]] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, PrimePowerQ[#] &], {n, 95}]

A317531(n) = sumdiv(n,d,((-1)^(n/d+1))*(1==omega(d))); \\ Antti Karttunen, Sep 30 2018

G.f.: Sum_{k>=1} x^A246655(k)/(1 + x^A246655(k)).
L.g.f.: log(Product_{p prime, k>=1} (1 + x^(p^k))^(1/p^k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A069513(d).
If n is odd, a(n) = A001222(n).

cp's OEIS Frontend

A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

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A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

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A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

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A106404 Number of even semiprimes dividing n.

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A205745 a(n) = card { d | d*p = n, d odd, p prime }.

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A373458 Expansion of Sum_{p prime} x^p/(1 - p*x^p).

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A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

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A305633 Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).

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