A316441 -id:A316441 - cp's OEIS frontend

A339721 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^6.

1, -6, -6, 15, -6, 30, -6, -26, 15, 30, -6, -60, -6, 30, 30, 51, -6, -60, -6, -60, 30, 30, -6, 96, 15, 30, -26, -60, -6, -114, -6, -102, 30, 30, 30, 96, -6, 30, 30, 96, -6, -114, -6, -60, -60, 30, -6, -210, 15, -60, 30, -60, -6, 96, 30, 96, 30, 30, -6, 156, -6, 30, -60, 172, 30

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022601, A316441, A339338, A339717, A339718, A339719, A339720, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339338(n/d) * a(d).

a(p^k) = A022601(k) for prime p.

A339722 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^7.

Original entry on oeis.org

1, -7, -7, 21, -7, 42, -7, -42, 21, 42, -7, -105, -7, 42, 42, 84, -7, -105, -7, -105, 42, 42, -7, 189, 21, 42, -42, -105, -7, -203, -7, -175, 42, 42, 42, 217, -7, 42, 42, 189, -7, -203, -7, -105, -105, 42, -7, -399, 21, -105, 42, -105, -7, 189, 42, 189, 42, 42, -7, 385

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022602, A316441, A339339, A339717, A339718, A339719, A339720, A339721.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339339(n/d) * a(d).

a(p^k) = A022602(k) for prime p.

A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, -1, 0, 2, 1, 1, 1, -2, 0, 1, 0, 0, 0, 2, 0, -2, -2, -1, 1, -1, -2, 3, -1, 1, -2, -3, -2, 3, 0, -3, 1, -4, -5, 1, -1, -2, -1, 5, -5, 1, -3, 1, -1, -5, -4, 5, 1, -1, -9, -2, -1, -6, -1, -3, -2, 7, -7, -8, -2, -2

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320836.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^(Length[m]-1),{m,mps[nrmptn[n]]}],{n,30}]

a(n) = A316441(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -2, -1, 0, -2, 0, -2, -1, -1, -1, -4, -1, -1, -1, -3, 0, -3, 0, -2, -4, -1, -1, -6, -2, -3, -2, -2, 0, -6, -2, -4, -1, -1, 0, -5, 0, -1, -3, -9, -2, -3, 0, -2, -1, -3, 0, -7, 0

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320835.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^Length[m],{m,Select[mps[nrmptn[n]],UnsameQ@@#&]}],{n,30}]

a(n) = A114592(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A328731 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^k.

Original entry on oeis.org

1, -2, -3, -1, -5, 0, -7, -4, -3, 0, -11, 3, -13, 0, 0, 3, -17, 6, -19, 5, 0, 0, -23, 18, -10, 0, -10, 7, -29, 30, -31, -2, 0, 0, 0, 24, -37, 0, 0, 30, -41, 42, -43, 11, 15, 0, -47, 27, -21, 20, 0, 13, -53, 38, 0, 42, 0, 0, -59, 60, -61, 0, 21, 17, 0, 66, -67, 17, 0, 70

Offset: 1

Ilya Gutkovskiy, Oct 26 2019

sign

Dirichlet inverse of A050368.

Cf. A006881 (positions of 0's), A050368, A316441, A328730.

a(1) = 1; a(n) = -Sum_{d|n, dA050368(n/d) * a(d).

A344369 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(1-s)).

Original entry on oeis.org

1, -2, -3, 0, -5, 0, -7, -8, 0, 0, -11, 0, -13, 0, 0, 16, -17, 0, -19, 0, 0, 0, -23, 24, 0, 0, -27, 0, -29, 30, -31, -32, 0, 0, 0, 36, -37, 0, 0, 40, -41, 42, -43, 0, 0, 0, -47, 0, 0, 0, 0, 0, -53, 54, 0, 56, 0, 0, -59, 60, -61, 0, 0, 64, 0, 66, -67, 0, 0, 70

Offset: 1

Ilya Gutkovskiy, May 16 2021

sign

Cf. A022693, A224892, A316441, A328731, A344368, A344370.

facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; A316441[n_] := Sum[(-1)^Length[f], {f, facs[n]}]; Table[n A316441[n], {n, 70}]

a(n) = n * A316441(n).

A328877 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k - 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 6, 10, 9, 13, 10, 22, 12, 19, 22, 25, 16, 36, 18, 40, 32, 31, 22, 69, 30, 37, 42, 58, 28, 89, 30, 70, 52, 49, 58, 121, 36, 55, 62, 125, 40, 129, 42, 94, 108, 67, 46, 203, 63, 115, 82, 112, 52, 174, 94, 181, 92, 85, 58, 319, 60, 91, 156, 182

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

nonn

Number of ways to factor n into distinct factors with 1 kind of 2, 2 kinds of 3, ..., k-1 kinds of k.

Dirichlet convolution of A050368 with A316441.

Cf. A050368, A316441, A328853, A328876.

seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, my(m=logint(n, k), p=(1 + x + O(x*x^m))^(k-1), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w)); v} \\ Andrew Howroyd, Oct 29 2019

a(n) = Sum_{d|n} A050368(n/d) * A316441(d).

A339734 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^8.

Original entry on oeis.org

1, -8, -8, 28, -8, 56, -8, -64, 28, 56, -8, -168, -8, 56, 56, 134, -8, -168, -8, -168, 56, 56, -8, 344, 28, 56, -64, -168, -8, -328, -8, -288, 56, 56, 56, 428, -8, 56, 56, 344, -8, -328, -8, -168, -168, 56, -8, -728, 28, -168, 56, -168, -8, 344, 56, 344, 56, 56, -8, 792

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A007259, A316441, A339340, A339717, A339718, A339719, A339720, A339721, A339722, A339735.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339340(n/d) * a(d).

a(p^k) = A007259(k) for prime p.

A339735 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^9.

Original entry on oeis.org

1, -9, -9, 36, -9, 72, -9, -93, 36, 72, -9, -252, -9, 72, 72, 207, -9, -252, -9, -252, 72, 72, -9, 585, 36, 72, -93, -252, -9, -495, -9, -459, 72, 72, 72, 765, -9, 72, 72, 585, -9, -495, -9, -252, -252, 72, -9, -1278, 36, -252, 72, -252, -9, 585, 72, 585, 72, 72, -9, 1449

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022604, A316441, A339341, A339717, A339718, A339719, A339720, A339721, A339722, A339734.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339341(n/d) * a(d).

a(p^k) = A022604(k) for prime p.

A351023 Dirichlet g.f.: Product_{k>=2} 1 / (1 + 2 * k^(-s)).

Original entry on oeis.org

1, -2, -2, 2, -2, 2, -2, -6, 2, 2, -2, -2, -2, 2, 2, 14, -2, -2, -2, -2, 2, 2, -2, 10, 2, 2, -6, -2, -2, 2, -2, -26, 2, 2, 2, 6, -2, 2, 2, 10, -2, 2, -2, -2, -2, 2, -2, -18, 2, -2, 2, -2, -2, 10, 2, 10, 2, 2, -2, 2, -2, 2, -2, 50, 2, 2, -2, -2, 2, 2, -2, -14, -2, 2, -2

Offset: 1

Ilya Gutkovskiy, Jan 29 2022

sign

Cf. A071109, A165872, A316441, A339717, A349921, A351024.

cp's OEIS Frontend

A339721 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^6.

Views

Author

Keywords

Crossrefs

Formula

A339722 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^7.

Views

Author

Keywords

Crossrefs

Formula

A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A328731 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^k.

Views

Author

Keywords

Comments

Crossrefs

Formula

A344369 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(1-s)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A328877 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k - 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A339734 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^8.

Views

Author

Keywords

Crossrefs

Formula

A339735 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^9.

Views

Author

Keywords

Crossrefs

Formula

A351023 Dirichlet g.f.: Product_{k>=2} 1 / (1 + 2 * k^(-s)).

Views

Author

Keywords

Crossrefs