A048995 -id:A048995 - cp's OEIS frontend

A174514 Partial sums of A048995.

1, 5, 56, 143, 238, 357, 480, 625, 786, 973, 1178, 1387, 1602, 1839, 2084, 2331, 2592, 2859, 3134, 3421, 3710, 4001, 4304, 4609, 4930, 5253, 5578, 5913, 6254, 6625, 7030, 7437, 7862, 8291, 8738, 9209, 9682, 10179, 10694, 11211, 11730, 12259, 12798, 13349, 13904, 14465, 15040, 15623, 16234, 16857, 17482, 18109, 18766

Offset: 1

Jonathan Vos Post, Nov 28 2010

nonn

The subsequence of values which are themselves in A048995 begins: 1, 625. The subsequence of primes begins 5, 4001, 8291, 9209.

			a(8) = 1 + 4 + 51 + 87 + 95 + 119 + 123 + 145 = 625 = 5^4.

Cf. A048050, A048995.

a(n) = SUM[i=1..n] A048995(i) = SUM[i=1..n] (numbers that are not the sum of the nontrivial factors, excluding 1 and i, of some natural number).

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

Original entry on oeis.org

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

A007369 Numbers n such that sigma(x) = n has no solution.

Original entry on oeis.org

2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

With an initial 1, may be constructed inductively in stages from the list L = {1,2,3,....} by the following sieve procedure. Stage 1. Add 1 as the first term of the sequence a(n) and strike off 1 from L. Stage n+1. Add the first (i.e. leftmost) term k of L as a new term of the sequence a(n) and strike off k, sigma(k), sigma(sigma(k)),.... from L. - Joseph L. Pe, May 08 2002
This sieve is a special case of a more general sieve. Let D be a subset of N and let f be an injection on D satisfying f(n) > n. Define the sieve process as follows: 1. Start with the empty sequence S and let E = D. 2. Append the smallest element s of E to S. 3. Remove s, f(s), f(f(s)), f(f(f(s))), ... from E. 4. Go to step 2. After this sieving process, S = D - f(D). To get the current sequence, take f = sigma and D = {n | n >= 2}. - Max Alekseyev, Aug 08 2005
By analogy with the untouchable numbers (A005114), these numbers could be named "sigma-untouchable". - Daniel Lignon, Mar 28 2014
The asymptotic density of this sequence is 1 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			a(4) = 10 because there is no x < 10 whose sigma(x) = 10.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Complement of A002191.
See A083532 for the gaps, i.e., first differences.
See A048995 for the missed sums of nontrivial divisors.

a = {}; Do[s = DivisorSigma[1, n]; a = Append[a, s], {n, 1, 115} ]; Complement[ Table[ n, {n, 1, 115} ], Union[a] ]

list(lim)=my(v=List(),u=vectorsmall(lim\1),t); for(n=1,lim, t=sigma(n); if(t<=lim, u[t]=1)); for(n=2,lim, if(u[n]==0, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Mar 09 2017

A007369_list(LIM,m=0,L=List(),s)={for(n=2,LIM,(s=sigma(n-1))>LIM || bittest(m,s) || m+=1<M. F. Hasler, Mar 12 2018

A175192(a(n)) = 0, A054973(a(n)) = 0. - Jaroslav Krizek, Mar 01 2010

a(n) < 2n + sqrt(8n). - Charles R Greathouse IV, Oct 23 2015

More terms from David W. Wilson

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56

Offset: 0

Labos Elemer, Apr 12 2002

For odd n >= 9, a(n) <= A073046((n-1)/2). - Max Alekseyev, Sep 01 2025

			For n=128: a(128)=16129, divisors={1,127,16129}, 1+127=sigma(n)-n=128 and 16129 is the smallest.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 9884 terms from Richard J. Mathar)
Mersenne Forum, Given sigma(n)-n, find the smallest possible n

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

See A359132 for another version.

f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1000000}]; t

a(n) = min(x, A001065(x)=n) or a(n)=0 if n is an untouchable number (i.e., if from A005114).

a(0)=1 prepended by Max Alekseyev, Sep 01 2025

A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

Original entry on oeis.org

0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56, 117

Offset: 1

Labos Elemer, Apr 12 2002

nonn

Remark that A070016(n)=A070015(n+1) in accordance with A048995(k)+1=A005114(k).

			n=127: a(n)=16129, divisors={1,127,16129}, 127=sigma[n]-n-1=127 and 16129 is the smallest.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

f1[x_] := DivisorSigma[1, x]-x-1; t=Table[0, {128}]; Do[b=f1[n]; If[b<129&&t[[b]]==0, t[[b]]=n], {n, 1, 1000000}]; t

a(n)=Min{x; A048050(x)=n} or a(n)=0 if n is from A048995.

A049030 Sum of sigma(j) for 1<=j<10^n, where sigma(j) = A048050(j) is the sum of the proper divisors >1 of j (excluding 1 and n).

Original entry on oeis.org

16, 3034, 320243, 32226805, 3224444759, 322465138002, 32246681892518, 3224670122682648, 322467031114802292, 32246703322412473945, 3224670334023621455211, 322467033422357645316809, 32246703342390510922780778, 3224670334240928188556405242

Offset: 1

Enoch Haga and Jud McCranie

			For n = 1, the sum of sigma(j), for j < 10 is 0 + 0 + 0 + 2 + 0 + 5 + 0 + 6 + 3 = 16, so a(1) = 16.

Amiram Eldar, Table of n, a(n) for n = 1..36 (calculated from the b-file at A049000)
Carlos Rivera, Problem 23.- Divisors (V) Aliquot sequences, Sociable Numbers, Prime Puzzles and Problems Connection.

Cf. A001065, A046915, A048050, A048995, A049000.

Cf. A072691 (Pi^2/12).

At a(3) = 320243, for example, take a(3) from A049000: 820741 - 500498 = 320243. Compute 500498 from 999*1000/2 = 499500, split evenly and reverse to 500499 - 1 = 500498. Add a 9 and 0 for each successive term.

a(n) = A049000(n) - 10^n * (10^n + 1) / 2 + 2 ~ (Pi^2/12 - 1/2) * 10^(2*n). - Amiram Eldar, Feb 16 2020

More terms from Amiram Eldar, Feb 16 2020

A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

Original entry on oeis.org

2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Impossible values for A048250 (numbers k in increasing order such that A048250(m) = k has no solution).
Numbers that are not of the form Product (p_i + 1), p is a prime, so all odd numbers (except 1 and 3) are in this sequence.
Also numbers that are not the sum of the divisors of some squarefree number.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A005114, A005117, A007369, A048250, A048995, A062822, A063948, A064000, A203444, A206778, A274793.

sort(convert({$1..1000} minus map(numtheory:-sigma, select(numtheory:-issqrfree, {$1..1000})),list)); # Robert Israel, Jun 26 2018

TakeWhile[Complement[Range@ #, Union@ Table[Total@ Select[Divisors@ n, SquareFreeQ], {n, 2 #}]], Function[k, k <= #]] &@ 111

cp's OEIS Frontend

A174514 Partial sums of A048995.

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

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A007369 Numbers n such that sigma(x) = n has no solution.

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A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

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A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

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A049030 Sum of sigma(j) for 1<=j<10^n, where sigma(j) = A048050(j) is the sum of the proper divisors >1 of j (excluding 1 and n).

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A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

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