A182936 -id:A182936 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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

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

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.

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

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

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

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

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

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 *)

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

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

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

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

A007956 Product of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, 1, 324, 1, 400, 21, 22, 1, 13824, 5, 26, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, 74088, 1, 1936, 2025, 46, 1, 5308416, 7, 2500, 51, 2704, 1, 157464, 55, 175616, 57, 58, 1, 777600000, 1, 62, 3969, 32768, 65

Offset: 1

R. Muller

From Bernard Schott, Feb 01 2019: (Start)
a(n) = 1 iff n = 1 or n is prime.
a(n) = n when n > 1 iff n has exactly four divisors, equally, iff n is either the cube of a prime or the product of two different primes, so iff n belongs to A030513 (very nice proof in Sierpiński).
a(p^3) = 1 * p * p^2 = p^3; a(p*q) = 1 * p * q = p*q.
As a(1) = 1, {1} Union A030513 = A007422, fixed points of this sequence. (End)

			a(18) = 1 * 2 * 3 * 6 * 9 = 324. - _Bernard Schott_, Jan 31 2019

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.
Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Zhu Weiyi, On the divisor product sequences, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.

Cf. A000005, A000720, A007955, A048671, A182936, A027751, A066729.
Cf. A007422 (fixed points). A030513 (subsequence).
Cf. A001065 (sums of proper divisors).

a007956 = product . a027751_row
-- Reinhard Zumkeller, Feb 04 2013, Nov 02 2011

A007956 := n -> mul(i,i=op(numtheory[divisors](n) minus {1,n}));
seq(A007956(i), i=1..79); # Peter Luschny, Mar 22 2011

Table[Times@@Most[Divisors[n]], {n, 65}] (* Alonso del Arte, Apr 18 2011 *)
a[n_] := n^(DivisorSigma[0, n]/2 - 1); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 07 2013 *)

A007956(n) = local(a);a=1;fordiv(n,d,a=a*d);a/n \\ Michael B. Porter, Dec 01 2009

a(n)=my(k); if(issquare(n, &k), k^(numdiv(n)-2), n^(numdiv(n)/2-1)) \\ Charles R Greathouse IV, Oct 15 2015

from math import isqrt
from sympy import divisor_count
def A007956(n): return isqrt(n)**(d-2) if (d:=divisor_count(n))&1 else n**((d>>1)-1) # Chai Wah Wu, Jun 18 2023

a(n) = A007955(n)/n = n^(A000005(n)/2-1) = sqrt(n^(number of factors of n other than 1 and n)).
a(n) = Product_{k=1..A000005(n)-1} A027751(n,k). - Reinhard Zumkeller, Feb 04 2013
a(n) = A240694(n, A000005(n)-1) for n > 1. - Reinhard Zumkeller, Apr 10 2014
Sum_{k=1..n} 1/a(k) ~ pi(n) + log(log(n))^2 + c_1*log(log(n)) + c_2 + O(log(log(n))/log(n)), where pi(n) = A000720(n) and c_1 and c_2 are constants (Weiyi, 2004; Sandor and Crstici, 2004). - Amiram Eldar, Oct 29 2022

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A048671 a(n) is the least common multiple of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 12, 1, 14, 15, 8, 1, 18, 1, 20, 21, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 16, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 7, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 32, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1

Offset: 1

Labos Elemer

A proper divisor d of n is a divisor of n such that 1 <= d < n.

Previous name was: a(n) = q(n)/q(n-1), where q(n) = n!/A003418(n).

			8!/lcm(8) = 48 = 40320/840 while 7!/lcm(7) = 5040/420 = 12 so a(8) = 48/12 = 4.
a(5) = 1 = lcm(1,2,3,4,5)/lcm(1,5,10,10,5,1).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
Index entries for sequences related to lcm's

Cf. A000142, A002944, A003418, A014963, A025527.

Cf. A182936 gives the dual (greatest common divisor).

A048671 := n -> ilcm(op(numtheory[divisors](n) minus {1,n}));
seq(A048671(i), i=1..79); # Peter Luschny, Mar 21 2011

{1}~Join~Table[LCM @@ Most@ Divisors@ n, {n, 2, 79}] (* Michael De Vlieger, May 01 2016 *)

a(n)=my(p=n);if(isprime(n)||(ispower(n,,&p)&&isprime(p)),n/p,n) \\ Charles R Greathouse IV, Jun 24 2011

a(n)=my(p); if(isprimepower(n,&p), n/p, n) \\ Charles R Greathouse IV, May 02 2016

def A048671(n) :
    if n < 2 : return 1
    else : D = divisors(n); D.pop()
    return lcm(D)
[A048671(i) for i in (1..79)] # Peter Luschny, Feb 03 2012

a(n) = A025527(n)/A025527(n-1).
a(n) = (n*A003418(n-1))/A003418(n).
a(n) = A003418(n-1)/A002944(n). [corrected by Michel Marcus, May 18 2020]
From Henry Bottomley, May 19 2000: (Start)
a(n) = n/A014963(n) = lcm(A052126(n), A032742(n)).
a(n) = n if n not a prime power, a(n) = n/p if n = p^m (i.e., a(n) = 1 if n = p). (End)
From Vladeta Jovovic, Jul 04 2002: (Start)
a(n) = n*Product_{d | n} d^mu(d).
Product_{d | n} a(d) = A007956(n). (End)
a(n) = Product_{k=1..n-1} if(gcd(n, k) > 1, 1 - exp(2*pi*i*k/n), 1), where i = sqrt(-1). - Paul Barry, Apr 15 2005
From Peter Luschny, Jun 09 2011: (Start)
a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*Pi/Gamma(k/n)^2, 1).
a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*sin(Pi*k/n), 1). (End)

New definition based on a comment of David Wasserman by Peter Luschny, Mar 23 2011

A380117 a(n) = n - A380118(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 3, 2, 0, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 3, 3, 4, 4, 0, 0, -2, -2, -1, -1, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, -3, -3, -3, -3, -2, -2, -2, -2, -2, -2, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3

Offset: 1

Peter Luschny, Jan 30 2025

sign

Conjecture: The sign of the terms changes infinitely often.

Peter Luschny, Table of n, a(n) for n = 1..20000

Cf. A380118, A182936.

ExpLambda := n -> n / ilcm(op(numtheory[divisors](n) minus {1, n})):
A380118 := proc(n) option remember; ifelse(n = 1, 1,
A380118(n-1) + ExpLambda(n) - ifelse(isprime(n), n, 0)) end:
a := n -> n - A380118(n): seq(a(n), n = 1..79);

A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 4, 6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 16, 16, 17, 18, 19, 19, 20, 25, 26, 29, 30, 30, 31, 31, 33, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 41, 42, 43, 44, 44, 45, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 60, 61, 61, 62, 63, 65, 66, 67, 67, 68, 69, 70

Offset: 1

Peter Luschny, Jan 30 2025

nonn

Cf. A014963, A061397, A048671, A010051, A182936, A034387, A072107, A380117.

pSum := L -> ListTools:-PartialSums(L): h := n -> n/A048671(n) - n*A010051(n):
aList := upto -> pSum([seq(h(k), k = 1..upto)]): aList(70);

Accumulate[Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0] , {n, 1, 70}]]

a(n) = A072107(n) - A034387(n). - Amiram Eldar, Jan 30 2025

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

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A007956 Product of the proper divisors of n.

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A048671 a(n) is the least common multiple of the proper divisors of n.

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A380117 a(n) = n - A380118(n).

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A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

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