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

1,4

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

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.

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

Example

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

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 *)

Prog

(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
(PARI) a(n)=if(n>1, sigma(n)-n-1, 0) \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
a048050 1 = 0
a048050 n = (subtract 1) $ sum $ a027751_row n
-- Reinhard Zumkeller, Feb 09 2013
(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

Crossrefs

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

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

Sequence in context: A104755 A242690 A054013 * A329375 A078153 A104035

Adjacent sequences: A048047 A048048 A048049 * A048051 A048052 A048053

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved