|
|
A062799
|
|
Inverse Möbius transform of the numbers of distinct prime factors (A001221).
|
|
25
|
|
|
0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 12, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 17, 1, 4, 7, 7, 4, 12, 1, 13, 4, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
Let us say that two divisors d_1 and d_2 of n are adjacent divisors if d_1/d_2 or d_2/d_1 is a prime. Then a(n) is the number of all pairs of adjacent divisors of n. - Vladimir Shevelev, Aug 16 2010
Equivalent to the preceding comment: a(n) is the number of edges in the directed multigraph on tau(n) vertices, vertices labeled by the divisors d_i of n, where edges connect vertex(d_i) and vertex(d_j) if the ratio of the labels is a prime. - R. J. Mathar, Sep 23 2011
Depends on the prime signature of n as follows: a(A025487(n)) = 0, 1, 2, 4, 3, 7, 4, 10, 12, 5, 12, 13, 20, 6, 17, 16, 28, 7, 22, 33, 19 ,32, 24, 36, 8, 27, 46, ... (n>=1). - R. J. Mathar, May 28 2017
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{d|n} A001221(d), that is, where d runs over divisors of n.
Let n=Product_{i=1..omega(n)} p(i)^e(i). a(n) = d[Product_{i=1..omega(n)} (1 + e(i)*x)]/dx|x=1. In other words, a(n) = Sum_{m>=1} A146289(n,m)*m. - Geoffrey Critzer, Feb 10 2015
G.f.: Sum_{k>=1} omega(k)*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s)^2*primezeta(s) where primezeta(s) = Sum_{prime p} p^(-s). - Benedict W. J. Irwin, Jul 16 2018
|
|
EXAMPLE
|
n = 255: divisors = {1, 3, 5, 15, 17, 51, 85, 255}, a(255) = 0+1+1+2+1+2+2+3 = 12.
|
|
MAPLE
|
read("transforms") ;
nops(numtheory[factorset](n)) ;
end proc:
omega := [seq(A001221(n), n=1..80)] ;
ones := [seq(1, n=1..80)] ;
N:= 1000: # to get a(1) to a(N)
B:= Vector(N, t-> nops(numtheory:-factorset(t))):
A:= Vector(N):
for d from 1 to N do
md:= d*[$1..floor(N/d)];
A[md]:= map(`+`, A[md], B[d])
od:
|
|
MATHEMATICA
|
f[n_] := Block[{d = Divisors[n], c = l = 0, k = 2}, l = Length[d]; While[k < l + 1, c = c + Length[ FactorInteger[ d[[k]] ]]; k++ ]; Return[c]]; Table[f[n], {n, 1, 100} ]
omega[n_] := Length[FactorInteger[n]]; SetAttributes[omega, Listable]; omega[1] := 0; A062799[n_] := Plus @@ omega[Divisors[n]] (* Enrique Pérez Herrero, Sep 08 2009 *)
|
|
PROG
|
(Haskell)
a062799 = sum . map a001221 . a027750_row
(PARI) a(n)=my(f=factor(n)[, 2], s); forvec(v=vector(#f, i, [0, f[i]]), s+=sum(i=1, #f, v[i]>0)); s \\ Charles R Greathouse IV, Oct 15 2015
(PARI) vector(100, n, sumdiv(n, k, omega(k))) \\ Altug Alkan, Oct 15 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|