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

Author

Ant King, Dec 21 2006

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Infinitary Divisor.

Crossrefs

Cf. A049417, A037445, A126164.

Programs

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]
f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); a[n_] := isigma[n] - n; Array[a, 100] (* Amiram Eldar, Mar 20 2025 *)

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.

Formula

a(n) = isigma(n) - n = A049417(n) - n.