A328745 -id:A328745 - cp's OEIS frontend

A329445 Dirichlet inverse of A328745.

1, -2, -3, 1, -5, 6, -7, 0, 3, 10, -11, -3, -13, 14, 15, 0, -17, -6, -19, -5, 21, 22, -23, 0, 10, 26, -1, -7, -29, -30, -31, 0, 33, 34, 35, 3, -37, 38, 39, 0, -41, -42, -43, -11, -15, 46, -47, 0, 21, -20, 51, -13, -53, 2, 55, 0, 57, 58, -59, 15, -61, 62, -21, 0, 65, -66, -67, -17

Offset: 1

Werner Schulte, Nov 13 2019

Signed version of A182938.

Cf. A008836, A182938, A328745.

from math import prod, comb
from sympy import factorint
def A329445(n): return prod(-comb(p,e) if e&1 else comb(p,e) for p,e in factorint(n).items()) # Chai Wah Wu, Dec 23 2022

Multiplicative with a(p^e) = (-1)^e*binomial(p,e) for prime p and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n)/n^s = Product_{p prime} (1-p^(-s))^p.
a(n) = A182938(n) * A008836(n) for n > 0.

A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 0, 3, 10, 11, 3, 13, 14, 15, 0, 17, 6, 19, 5, 21, 22, 23, 0, 10, 26, 1, 7, 29, 30, 31, 0, 33, 34, 35, 3, 37, 38, 39, 0, 41, 42, 43, 11, 15, 46, 47, 0, 21, 20, 51, 13, 53, 2, 55, 0, 57, 58, 59, 15, 61, 62, 21, 0, 65, 66

Offset: 1

Peter Luschny, Jan 16 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.

Cf. A027748, A124010, A007318, A328745.

a182938 n = product $ zipWith a007318'
   (a027748_row n) (map toInteger $ a124010_row n)
-- Reinhard Zumkeller, Feb 18 2012

A182938 := proc(n) local e,j; e := ifactors(n)[2]:
mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
seq (A182938(n), n=1..100);

a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
Table[a[n], {n, 1, 100}] (* Kellen Myers, Jan 16 2011 *)

a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ M. F. Hasler

for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ Vaclav Kotesovec, Mar 28 2025

a(A185359(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - Ilya Gutkovskiy, Oct 26 2019
Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - Vaclav Kotesovec, Mar 28 2025

Given terms checked with new PARI code by M. F. Hasler, Jan 16 2011

cp's OEIS Frontend

A329445 Dirichlet inverse of A328745.

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A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

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