OFFSET
1,6
COMMENTS
A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.
FORMULA
a(n) = isigma(n) - n = A049417(n)-n.
EXAMPLE
As the infinitary divisors of 240 are 1, 3, 5, 15, 16, 48, 80, 240, we have a(240) = 1 + 3 + 5 + 15 + 16 + 48 + 80 = 168.
MAPLE
A049417 := proc(n)
local a, pe, k, edgs, p ;
a := 1;
for pe in ifactors(n)[2] do
p := op(1, pe) ;
edgs := convert(op(2, pe), base, 2) ;
for k from 0 to nops(edgs)-1 do
dk := op(k+1, edgs) ;
a := a*(p^(2^k*(1+dk))-1)/(p^(2^k)-1) ;
end do:
end do:
a ;
end proc:
A126168 := proc(n)
A049417(n)-n ;
end proc:
seq(A126168(n), n=1..100) ; # R. J. Mathar, Jul 23 2021
MATHEMATICA
ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, # ]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer?Positive] := Module[ { factors = First /@ FactorInteger[n], d = Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][ #, Last[ # ]]] & /@ Transpose[Last /@ ExponentList[ #, factors] & /@ d]], _?( And @@ # &), {1}]] ]] ] Null; properinfinitarydivisorsum[k_] := Plus @@ InfinitaryDivisors[k] - k; properinfinitarydivisorsum /@ Range[75]
PROG
(PARI)
A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))} \\ This function from Andrew Lelechenko, Apr 22 2014
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ant King, Dec 21 2006
STATUS
approved