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A126168 Sum of the proper infinitary divisors of n.

%I A126168 #15 Mar 20 2025 03:35:41
%S A126168 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,
%T A126168 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,
%U A126168 17,64,23,32,1,60,1,34,17,21,19,78,1,22,27,74,1,78,1,40,29
%N A126168 Sum of the proper infinitary divisors of n.
%C A126168 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.
%H A126168 Antti Karttunen, <a href="/A126168/b126168.txt">Table of n, a(n) for n = 1..10000</a>
%H A126168 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>.
%F A126168 a(n) = isigma(n) - n = A049417(n) - n.
%e A126168 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.
%p A126168 A049417 := proc(n)
%p A126168     local a,pe,k,edgs,p ;
%p A126168     a := 1;
%p A126168     for pe in ifactors(n)[2] do
%p A126168         p := op(1,pe) ;
%p A126168         edgs := convert(op(2,pe),base,2) ;
%p A126168         for k from 0 to nops(edgs)-1 do
%p A126168             dk := op(k+1,edgs) ;
%p A126168             a := a*(p^(2^k*(1+dk))-1)/(p^(2^k)-1) ;
%p A126168         end do:
%p A126168     end do:
%p A126168     a ;
%p A126168 end proc:
%p A126168 A126168 := proc(n)
%p A126168     A049417(n)-n ;
%p A126168 end proc:
%p A126168 seq(A126168(n),n=1..100) ; # _R. J. Mathar_, Jul 23 2021
%t A126168 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]
%t A126168 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 *)
%o A126168 (PARI)
%o A126168 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
%o A126168 A126168(n) = (A049417(n) - n); \\ _Antti Karttunen_, Oct 04 2017, after the given formula.
%Y A126168 Cf. A049417, A037445, A126164.
%K A126168 easy,nonn
%O A126168 1,6
%A A126168 _Ant King_, Dec 21 2006