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A126168
Sum of the proper infinitary divisors of n.
35
0, 1, 1, 1, 1, 6, 1, 7, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 36, 1, 16, 13, 12, 1, 42, 1, 19, 15, 20, 13, 14, 1, 22, 17, 50, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 66, 17, 64, 23, 32, 1, 60, 1, 34, 17, 21, 19, 78, 1, 22, 27, 74, 1, 78, 1, 40, 29
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
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
A126168(n) = (A049417(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula.
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ant King, Dec 21 2006
STATUS
approved