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A131902 Smallest positive integer k with the same sum of divisors as the n-th integer for which such a k exists.

%I A131902 #16 Apr 08 2025 08:47:40
%S A131902 6,14,10,14,16,20,21,33,24,28,20,30,33,30,34,30,54,40,24,42,44,42,66,
%T A131902 30,48,42,60,57,68,44,54,40,60,66,54,52,63,85,102,74,66,104,88,66,80,
%U A131902 60,84,99,93,96,86,114,76,132,105,102,60,88,111,90,138,105,114,102,105,138,96
%N A131902 Smallest positive integer k with the same sum of divisors as the n-th integer for which such a k exists.
%H A131902 Robert Israel, <a href="/A131902/b131902.txt">Table of n, a(n) for n = 1..10000</a>
%H A131902 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%F A131902 Let S = {n>0 : there exists a k>0 and k<n with sigma(k) = sigma(n)}. Then a(n) := min(k>0: sigma(k) = sigma(n-th element of S)).
%e A131902 a(3)=10 because 17 is the third integer for which a smaller integer with same sum of divisors exists and sigma(17)=1+17=18 and sigma(10)=1+2+5+10=18 and there is no k>0 less than 10 with sigma(k)=18.
%p A131902 N:= 1000: # to use values of sigma <= N
%p A131902 V:= Vector(N): A:= Vector(N):
%p A131902 for n from 1 to N do
%p A131902   v:= numtheory:-sigma(n);
%p A131902   if v <= N then
%p A131902     if V[v] = 0 then V[v]:= n
%p A131902     else A[n]:= V[v]
%p A131902     fi
%p A131902   fi
%p A131902 od:
%p A131902 subs(0=NULL, convert(A,list)); # _Robert Israel_, Mar 30 2018
%t A131902 Clear[tmp]; Function[n,If[Head[ #1]===tmp,#1=n;Unevaluated[Sequence[]],#1]& [tmp[DivisorSigma[1,n]]]]/@Range[200]
%o A131902 (PARI) list(lim) = my(m); for(k = 1, lim, m = invsigmaMin(sigma(k)); if(m < k, print1(m, ", "))); \\ _Amiram Eldar_, Dec 20 2024, using _Max Alekseyev_'s invphi.gp
%Y A131902 Cf. A069822, A131903, A131904, A131905, A131906, A131907, A131908.
%K A131902 easy,nonn
%O A131902 1,1
%A A131902 Peter Pein (petsie(AT)dordos.net), Jul 26 2007