A048050 Chowla's function: sum of divisors of n except for 1 and n.

0, 0, 0, 2, 0, 5, 0, 6, 3, 7, 0, 15, 0, 9, 8, 14, 0, 20, 0, 21, 10, 13, 0, 35, 5, 15, 12, 27, 0, 41, 0, 30, 14, 19, 12, 54, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 75, 7, 42, 20, 45, 0, 65, 16, 63, 22, 31, 0, 107, 0, 33, 40, 62, 18, 77, 0, 57, 26, 73, 0, 122, 0, 39, 48, 63, 18, 89

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

a(n) = 0 if and only if n is a noncomposite number (cf. A008578). - Omar E. Pol, Jul 31 2012

If n is semiprime, a(n) = A008472(n). - Wesley Ivan Hurt, Aug 22 2013

If n = p*q where p and q are distinct primes then a(n) = p+q.

If k,m > 1 are coprime, then a(k*m) = a(k)*a(m) + (m+1)*a(k) + (k+1)*a(m) + k + m. - Robert Israel, Apr 28 2015

a(n) is also the total number of parts in the partitions of n into equal parts that contain neither 1 nor n as a part (see example). More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that contain neither k nor k*n as a part. - Omar E. Pol, Nov 24 2019

Named after the Indian-American mathematician Sarvadaman D. S. Chowla (1907-1995). - Amiram Eldar, Mar 09 2024

Examples

			For n = 20 the divisors of 20 are 1,2,4,5,10,20, so a(20) = 2+4+5+10 = 21.
On the other hand, the partitions of 20 into equal parts that contain neither 1 nor 20 as a part are [10,10], [5,5,5,5], [4,4,4,4,4], [2,2,2,2,2,2,2,2,2,2]. There are 21 parts, so a(20) = 21. - _Omar E. Pol_, Nov 24 2019
		

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., 25 (1971), 923-925.
• Abdur Rahman Nasir, On a certain arithmetic function, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
• Omar E. Pol, Illustration of initial terms obtained geometrically.

Crossrefs

Cf. A000203, A001065, A000593, A002954, A048995, A007956, A048671, A182936.

Cf. A057533, A005276, A027751, A237593, A245092, A244049 (partial sums).

Programs

Haskell

a048050 1 = 0
a048050 n = (subtract 1) $ sum $ a027751_row n
-- Reinhard Zumkeller, Feb 09 2013

Magma

A048050:=func< n | n eq 1 or IsPrime(n) select 0 else &+[ a: a in Divisors(n) | a ne 1 and a ne n ] >; [ A048050(n): n in [1..100] ]; // Klaus Brockhaus, Mar 04 2011

Maple

A048050 := proc(n) if n > 1 then numtheory[sigma](n)-1-n ; else 0; end if; end proc:

Mathematica

f[n_]:=Plus@@Divisors[n]-n-1; Table[f[n],{n,100}] (*Vladimir Joseph Stephan Orlovsky, Sep 13 2009*)
Join[{0},DivisorSigma[1,#]-#-1&/@Range[2,80]] (* Harvey P. Dale, Feb 25 2015 *)

PARI

a(n)=if(n>1,sigma(n)-n-1,0) \\ Charles R Greathouse IV, Nov 20 2012

Python

from sympy import divisors
def a(n): return sum(divisors(n)[1:-1]) # Indranil Ghosh, Apr 26 2017

Python

from sympy import divisor_sigma
def A048050(n): return 0 if n == 1 else divisor_sigma(n)-n-1 # Chai Wah Wu, Apr 18 2021

Formula

a(n) = A000203(n) - A065475(n).

a(n) = A001065(n) - 1, n > 1.

For n > 1: a(n) = Sum_{k=2..A000005(n)-1} A027750(n,k). - Reinhard Zumkeller, Feb 09 2013

a(n) = A000203(n) - n - 1, n > 1. - Wesley Ivan Hurt, Aug 22 2013

G.f.: Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017