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
a(A001248(n)) = 2. - Reinhard Zumkeller, Dec 02 2014
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
T. D. Noe, Table of n, a(n) for n = 1..10000
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
E. Pérez Herrero, Psychedelic Geometry Blogspot, CURIOUS SERIES-002
FORMULA
a(n) = Sum_{d|n} A001221(d), that is, where d runs over divisors of n.
For squarefree s (i.e., s in A005117), a(s) = omega(s)*2^(omega(s)-1), where omega(n) = A001221(n). Also, for n>1, a(n) <= omega(n)*A000005(n) - 1. - Enrique Pérez Herrero, Sep 08 2009
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
a(A000040(n)) = 1; a(A001248(n)) = 2; a(A030078(n)) = 3; a(A030514(n)) = 4; a(A050997(n)) = 5. - Altug Alkan, Oct 17 2015
a(n) = Sum_{prime p|n} A000005(n/p). - Max Alekseyev, Aug 11 2016
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") ;
A001221 := proc(n)
nops(numtheory[factorset](n)) ;
end proc:
omega := [seq(A001221(n), n=1..80)] ;
ones := [seq(1, n=1..80)] ;
DIRICHLET(ones, omega) ; # R. J. Mathar, Sep 23 2011
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:
convert(A, list); # Robert Israel, Oct 21 2015
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
-- Reinhard Zumkeller, Dec 02 2014
(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
Labos Elemer, Jul 19 2001
STATUS
approved