A074753 Number of integers k such that sigma(k) < n.

0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 6, 6, 8, 9, 10, 11, 11, 11, 13, 13, 14, 14, 14, 14, 17, 17, 17, 17, 18, 18, 19, 21, 23, 23, 23, 23, 24, 24, 25, 26, 27, 27, 30, 30, 31, 31, 31, 31, 34, 34, 34, 34, 34, 34, 36, 36, 38, 39, 39, 39, 42, 42, 43, 44, 44, 44, 44, 44, 45, 45, 45, 45, 50, 50

Offset: 1

Author

Benoit Cloitre, Sep 28 2002

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Robert E. Dressler, An elementary proof of a theorem of Erdős on the sum of divisors function, Journal of Number Theory, Vol. 4, No. 6 (1972), pp. 532-536.

Crossrefs

Partial sums of A054973.

Cf. A000203, A308039.

Programs

Maple

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
for n from 1 to N-2 do
  s:= numtheory:-sigma(n)+1;
  if s <= N then V[s]:= V[s]+1 fi;
od:
ListTools:-PartialSums(V); # Robert Israel, Jan 08 2018

Mathematica

Table[Length[Select[Range[n], DivisorSigma[1,#] < n&]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *)

PARI

a(n)=sum(i=1,n,if(1+sign(sigma(i)-n),0,1))

PARI

list(nmax) = my(s = 0); for(n = 1, nmax, s += invsigmaNum(n); print1(s, ", ")); \\ Amiram Eldar, Dec 23 2024, using Max Alekseyev's invphi.gp

Formula

a(n) = card( k : sigma(k) < n ).

a(n) is asymptotic to c*n with c = 0.67...

a(n) = c * n + o(n), where c = 0.6727383... = A308039 (Dressler, 1972). - Amiram Eldar, Dec 23 2024