A126168 Sum of the proper infinitary divisors of n.

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

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
A126168(n) = (A049417(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula.

Crossrefs

Cf. A049417, A037445, A126164.

Sequence in context: A076714 A299620 A113811 * A331970 A028323 A137235

Adjacent sequences: A126165 A126166 A126167 * A126169 A126170 A126171

Keyword

easy,nonn

Author

Ant King, Dec 21 2006

Status

approved