|
%I #105 Mar 09 2024 04:42:43
%S 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,
%T 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,
%U 16,63,22,31,0,107,0,33,40,62,18,77,0,57,26,73,0,122,0,39,48,63,18,89
%N Chowla's function: sum of divisors of n except for 1 and n.
%C a(n) = 0 if and only if n is a noncomposite number (cf. A008578). - _Omar E. Pol_, Jul 31 2012
%C If n is semiprime, a(n) = A008472(n). - _Wesley Ivan Hurt_, Aug 22 2013
%C If n = p*q where p and q are distinct primes then a(n) = p+q.
%C 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
%C 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
%C Named after the Indian-American mathematician Sarvadaman D. S. Chowla (1907-1995). - _Amiram Eldar_, Mar 09 2024
%H T. D. Noe, <a href="/A048050/b048050.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Lal and A. Forbes, <a href="http://dx.doi.org/10.1090/S0025-5718-1971-0297685-X">A note on Chowla's function</a>, Math. Comp., 25 (1971), 923-925.
%H Abdur Rahman Nasir, <a href="https://archive.org/details/dli.calcutta.11605/page/140/mode/2up">On a certain arithmetic function</a>, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
%H Omar E. Pol, <a href="/A048050/a048050.jpg">Illustration of initial terms obtained geometrically</a>.
%F a(n) = A000203(n) - A065475(n).
%F a(n) = A001065(n) - 1, n > 1.
%F For n > 1: a(n) = Sum_{k=2..A000005(n)-1} A027750(n,k). - _Reinhard Zumkeller_, Feb 09 2013
%F a(n) = A000203(n) - n - 1, n > 1. - _Wesley Ivan Hurt_, Aug 22 2013
%F G.f.: Sum_{k>=2} k*x^(2*k)/(1 - x^k). - _Ilya Gutkovskiy_, Jan 22 2017
%e For n = 20 the divisors of 20 are 1,2,4,5,10,20, so a(20) = 2+4+5+10 = 21.
%e 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
%p A048050 := proc(n) if n > 1 then numtheory[sigma](n)-1-n ; else 0; end if; end proc:
%t f[n_]:=Plus@@Divisors[n]-n-1; Table[f[n],{n,100}] (*_Vladimir Joseph Stephan Orlovsky_, Sep 13 2009*)
%t Join[{0},DivisorSigma[1,#]-#-1&/@Range[2,80]] (* _Harvey P. Dale_, Feb 25 2015 *)
%o (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
%o (PARI) a(n)=if(n>1,sigma(n)-n-1,0) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Haskell)
%o a048050 1 = 0
%o a048050 n = (subtract 1) $ sum $ a027751_row n
%o -- _Reinhard Zumkeller_, Feb 09 2013
%o (Python)
%o from sympy import divisors
%o def a(n): return sum(divisors(n)[1:-1]) # _Indranil Ghosh_, Apr 26 2017
%o (Python)
%o from sympy import divisor_sigma
%o def A048050(n): return 0 if n == 1 else divisor_sigma(n)-n-1 # _Chai Wah Wu_, Apr 18 2021
%Y Cf. A000203, A001065, A000593, A002954, A048995, A007956, A048671, A182936.
%Y Cf. A057533, A005276, A027751, A237593, A245092, A244049 (partial sums).
%K nonn,nice,easy
%O 1,4
%A _N. J. A. Sloane_
|
