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

6, 14, 10, 14, 16, 20, 21, 33, 24, 28, 20, 30, 33, 30, 34, 30, 54, 40, 24, 42, 44, 42, 66, 30, 48, 42, 60, 57, 68, 44, 54, 40, 60, 66, 54, 52, 63, 85, 102, 74, 66, 104, 88, 66, 80, 60, 84, 99, 93, 96, 86, 114, 76, 132, 105, 102, 60, 88, 111, 90, 138, 105, 114, 102, 105, 138, 96

Offset: 1

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

• easy
• nonn

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A069822, A131903, A131904, A131905, A131906, A131907, A131908.

Programs

Maple

N:= 1000: # to use values of sigma <= N
V:= Vector(N): A:= Vector(N):
for n from 1 to N do
  v:= numtheory:-sigma(n);
  if v <= N then
    if V[v] = 0 then V[v]:= n
    else A[n]:= V[v]
    fi
  fi
od:
subs(0=NULL, convert(A,list)); # Robert Israel, Mar 30 2018

Mathematica

Clear[tmp]; Function[n,If[Head[ #1]===tmp,#1=n;Unevaluated[Sequence[]],#1]& [tmp[DivisorSigma[1,n]]]]/@Range[200]

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

Formula

Let S = {n>0 : there exists a k>0 and k0: sigma(k) = sigma(n-th element of S)).