A296075 -id:A296075 - cp's OEIS frontend

A378532 Dirichlet convolution of A296075 and A378525.

1, 0, 0, -2, 0, -3, 0, -2, -2, -3, 0, -4, 0, -3, -3, -2, 0, -4, 0, -4, -3, -3, 0, -2, -2, -3, -2, -4, 0, -6, 0, -2, -3, -3, -3, -1, 0, -3, -3, -2, 0, -6, 0, -4, -4, -3, 0, -2, -2, -4, -3, -4, 0, -2, -3, -2, -3, -3, 0, 0, 0, -3, -4, -2, -3, -6, 0, -4, -3, -6, 0, 0, 0, -3, -4, -4, -3, -6, 0, -2, -2, -3, 0, 0, -3, -3, -3, -2

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Inverse Möbius transform of A378534.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033879, A296075, A378531 (Dirichlet inverse), A378534 (Möbius transform), A378525, A378542.

Cf. also A378218.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
memoA378525 = Map();
A378525(n) = if(1==n,1,my(v); if(mapisdefined(memoA378525,n,&v), v, v = -sumdiv(n,d,if(dA378542(n/d)*A378525(d),0)); mapput(memoA378525,n,v); (v)));
A378532(n) = sumdiv(n,d,A296075(d)*A378525(n/d));

a(n) = Sum_{d|n} A296075(d)*A378525(n/d).

a(n) = Sum_{d|n} A378534(d).

A378432 Dirichlet inverse of A296075, where A296075 is the sum of deficiencies of divisors of n.

Original entry on oeis.org

1, -2, -3, 1, -5, 8, -7, 0, 1, 12, -11, -3, -13, 16, 17, 0, -17, -4, -19, -5, 23, 24, -23, 2, 1, 28, -1, -7, -29, -44, -31, 0, 35, 36, 37, 5, -37, 40, 41, 2, -41, -60, -43, -11, -7, 48, -47, 4, 1, -8, 53, -13, -53, 0, 57, 2, 59, 60, -59, 25, -61, 64, -9, 0, 67, -92, -67, -17, 71, -92, -71, 6, -73, 76, -9, -19, 79

Offset: 1

Antti Karttunen, Nov 26 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A033879.
Dirichlet inverse of A296075.
Möbius transform of A323910.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
memoA378432 = Map();
A378432(n) = if(1==n,1,my(v); if(mapisdefined(memoA378432,n,&v), v, v = -sumdiv(n,d,if(dA296075(n/d)*A378432(d),0)); mapput(memoA378432,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA296075(n/d) * a(d).

A033879 Deficiency of n, or 2n - (sum of divisors of n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, -4, 12, 4, 6, 1, 16, -3, 18, -2, 10, 8, 22, -12, 19, 10, 14, 0, 28, -12, 30, 1, 18, 14, 22, -19, 36, 16, 22, -10, 40, -12, 42, 4, 12, 20, 46, -28, 41, 7, 30, 6, 52, -12, 38, -8, 34, 26, 58, -48, 60, 28, 22, 1, 46, -12, 66, 10, 42, -4, 70, -51

Offset: 1

N. J. A. Sloane

Records for the sequence of the absolute values are in A075728 and the indices of these records in A074918. - R. J. Mathar, Mar 02 2007
a(n) = 1 iff n is a power of 2. a(n) = n - 1 iff n is prime. - Omar E. Pol, Jan 30 2014
If a(n) = 1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079. - Jianing Song, Oct 13 2019
It is not known whether there are any -1's in this sequence. See comment in A033880. - Antti Karttunen, Feb 02 2020

			For n = 10 the divisors of 10 are 1, 2, 5, 10, so the deficiency of 10 is 10 minus the sum of its proper divisors or simply 10 - 5 - 2 - 1 = 2. - _Omar E. Pol_, Dec 27 2013

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

N. J. A. Sloane, Table of n, a(n) for n = 1..25000 [First 2000 terms from T. D. Noe, terms up to 16384 from Antti Karttunen]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1610.01868 [math.NT], 2016.
Jose Arnaldo B. Dris, Analysis of the Ratio D(n)/n, arXiv:1703.09077 [math.NT], 2017.
Jose Arnaldo Bebita Dris, On a curious biconditional involving the divisors of odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23(4) (2017), 1-13.
Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego, Some Modular Considerations Regarding Odd Perfect Numbers, arXiv:2002.12139 [math.NT], 2020.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, Conditions equivalent to the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers - Part II, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 62-67.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Index entries for sequences related to sigma(n).

Cf. A000396 (positions of zeros), A005100 (of positive terms), A005101 (of negative terms).
Cf. A000203, A033880, A074918, A075728, A192895, A286385, A286449, A295881, A295882.
Cf. A083254 (Möbius transform), A228058, A296074, A296075, A323910, A325636, A325826, A325970, A325976.
Cf. A141545 (positions of a(n) = -12).
For this sequence applied to various permutations of natural numbers and some other sequences, see A323174, A323244, A324055, A324185, A324546, A324574, A324575, A324654, A325379.

with(numtheory): A033879:=n->2*n-sigma(n): seq(A033879(n), n=1..100);

Table[2n-DivisorSigma[1,n],{n,80}] (* Harvey P. Dale, Oct 24 2011 *)

a(n)=2*n-sigma(n) \\ Charles R Greathouse IV, Oct 13 2016

from sympy import divisor_sigma
def A033879(n): return (n<<1)-divisor_sigma(n) # Chai Wah Wu, Apr 13 2024

[2*n-sigma(n, 1) for n in range(1, 73)] # Stefano Spezia, Jul 18 2025

a(n) = -A033880(n).
a(n) = A005843(n) - A000203(n). - Omar E. Pol, Dec 14 2008
a(n) = n - A001065(n). - Omar E. Pol, Dec 27 2013
G.f.: 2*x/(1 - x)^2 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 24 2017
a(n) = A286385(n) - A252748(n). - Antti Karttunen, May 13 2017
From Antti Karttunen, Dec 29 2017: (Start)
a(n) = Sum_{d|n} A083254(d).
a(n) = Sum_{d|n} A008683(n/d)*A296075(d).
a(n) = A065620(A295881(n)) = A117966(A295882(n)).
a(n) = A294898(n) + A000120(n).
(End)
From Antti Karttunen, Jun 03 2019: (Start)
Sequence can be represented in arbitrarily many ways as a difference of the form (n - f(n)) - (g(n) - n), where f and g are any two sequences whose sum f(n)+g(n) = sigma(n). Here are few examples:
a(n) = A325314(n) - A325313(n) = A325814(n) - A034460(n) = A325978(n) - A325977(n).
a(n) = A325976(n) - A325826(n) = A325959(n) - A325969(n) = A003958(n) - A324044(n).
a(n) = A326049(n) - A326050(n) = A326055(n) - A326054(n) = A326044(n) - A326045(n).
a(n) = A326058(n) - A326059(n) = A326068(n) - A326067(n).
a(n) = A326128(n) - A326127(n) = A066503(n) - A326143(n).
a(n) = A318878(n) - A318879(n).
a(A228058(n)) = A325379(n). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Pi^2/12 = 0.177532... . - Amiram Eldar, Dec 07 2023

Definition corrected by N. J. A. Sloane, Jul 04 2005

A066218 Numbers k such that sigma(k) = Sum_{j|k, j

Original entry on oeis.org

198, 608, 11322, 20826, 56608, 3055150, 565344850, 579667086, 907521650, 8582999958, 13876688358, 19244570848, 195485816050, 255701999358, 1038635009650, 1410759512050, 3308222326688, 6293446033554, 12859914783762, 15343909268584, 18359652610976, 19142664182226, 41584649258178, 45090324794034, 56293124233554

Offset: 1

Joseph L. Pe, Dec 17 2001

I propose this generalization of perfect numbers: for an arithmetical function f, the "f-perfect numbers" are the n such that f(n) = sum of f(k) where k ranges over proper divisors of n. The usual perfect numbers are i-perfect numbers, where i is the identity function. This sequence lists the sigma-perfect numbers. It is not hard to see that the EulerPhi-perfect numbers are the powers of 2 and the d-perfect numbers are the squares of primes (d(n) denotes the number of divisors of n).
Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.
a(17) > 2*10^12. - Giovanni Resta, Jun 20 2013
Numbers k such that A296075(k) = 0. - Amiram Eldar, Apr 16 2024
No more terms < 10^14. - Jud McCranie, Nov 28 2024

			Proper divisors of 198 = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99}; sum of their sigma values = 1 + 3 + 4 + 12 + 13 + 12 + 39 + 36 + 48 + 144 + 156 = 468 = sigma(198).

Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Giovanni Resta, 34 numbers > 3*10^12 which belong to the sequence.

Cf. A211779, A000203, A066229, A066230, A296075.

f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

is(n)=sumdiv(n,d,sigma(d))==2*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Integer n = p1^k1 * p2^k2 * ... * pm^km is in this sequence if and only if g(p1^k1)*g(p2^k2)*...*g(pm^km)=2, where g(p^k) = (p^(k+2)-(k+2)*p+k+1)/(p^(k+1)-1)/(p-1) for prime p and integer k>=1. - Max Alekseyev, Oct 23 2008

More terms from Naohiro Nomoto, May 07 2002
a(7)-a(8) from Farideh Firoozbakht, Sep 18 2006
a(9)-a(13) from Donovan Johnson, Jun 25 2012
a(14)-a(16) from Giovanni Resta, Jun 20 2013
a(17)-a(25) from Jud McCranie, Nov 28 2024

A296074 Sum of deficiencies of the proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 6, 1, 5, 1, 8, 7, 4, 1, 9, 1, 9, 9, 12, 1, 2, 5, 14, 8, 13, 1, 16, 1, 5, 13, 18, 11, 3, 1, 20, 15, 8, 1, 24, 1, 21, 18, 24, 1, -9, 7, 27, 19, 25, 1, 20, 15, 14, 21, 30, 1, -1, 1, 32, 24, 6, 17, 40, 1, 33, 25, 40, 1, -27, 1, 38, 32, 37, 17, 48, 1, -1, 22, 42, 1, 9, 21, 44, 31, 26, 1, 18, 19, 45, 33, 48, 23, -36, 1, 53, 36, 33

Offset: 1

Antti Karttunen, Dec 04 2017

			For n = 6, whose proper divisors are 1, 2, 3, their deficiencies are 1, 1, 2, thus a(6) = 1+1+2 = 4.
For n = 12, whose proper divisors are 1, 2, 3, 4, 6, their deficiencies are 1, 1, 2, 1, 0, thus a(12) = 1+1+2+1+0 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A033879.

Cf. also A294886, A294887, A294888, A294889, A293438 (product of).

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2; a[1] = 0; a[n_] := Module[{f = FactorInteger[n]}, 3 * Times @@ f1 @@@ f - Times @@ f2 @@@ f - 2*n]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A033879(n) = ((2*n)-sigma(n));
A296074(n) = sumdiv(n,d,(dA033879(d));

a(n) = Sum_{d|n, dA033879(d).
a(n) = A296075(n) - A033879(n).
Sum_{k=1..n} a(k) ~ (Pi^2/4 - Pi^4/72 - 1) * n^2. - Amiram Eldar, Dec 04 2023

A318679 Sum of abundancies of abundant divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 16, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 26, 0, 0, 0, 12, 0, 12, 0, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 15, 0, 8, 0, 0, 0, 66, 0, 0, 0, 0, 0, 12, 0, 0, 0, 4, 0, 89, 0, 0, 0, 0, 0, 12, 0, 38, 0, 0, 0, 72, 0, 0, 0, 4, 0, 69, 0, 0, 0, 0, 0, 104, 0, 0, 0, 19, 0, 12, 0, 2, 0

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A033880, A187795, A296075, A318441, A318442, A318678.

f[n_] := DivisorSigma[1, n] - 2 n; Array[DivisorSum[#, f, f@ # >= 0 &] &, 105] (* Michael De Vlieger, Sep 04 2018 *)

A318679(n) = sumdiv(n,d,d=sigma(d)-(2*d); (d>0)*d);

a(n) = Sum_{d|n} [A033880(d) > 0]*A033880(d).

a(n) = A318678(n) - A296075(n).

A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 2, 7, -8, 9, 4, 4, 0, 14, -4, 15, -2, 10, 12, 18, -22, 18, 16, 13, 1, 24, -14, 25, 0, 23, 26, 24, -31, 33, 28, 27, -14, 37, -6, 38, 13, 15, 34, 41, -52, 41, 22, 40, 19, 48, -10, 42, -10, 45, 46, 53, -76, 55, 48, 29, 0, 55, 12, 63, 34, 57, 18, 66, -98, 69, 64, 42, 37, 64, 16, 73, -42, 51, 72, 78, -74, 74, 74, 73, 6

Offset: 1

Antti Karttunen, Aug 27 2018

sign

Inverse Möbius transform of A294898.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A005187, A007429, A093653, A294898, A296075, A318446, A318447.

Cf. also A297114, A317844.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318448(n) = sumdiv(n,d,A294898(d));

a(n) = Sum_{d|n} A294898(d).
a(n) = A318447(n) + A294898(n).
a(n) = A318446(n) - A007429(n).
a(n) = A296075(n) - A093653(n).

A318678 Sum of deficiencies of deficient divisors of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 7, 4, 8, 8, 11, 5, 13, 12, 13, 5, 17, 9, 19, 9, 19, 20, 23, 6, 24, 24, 22, 13, 29, 16, 31, 6, 31, 32, 33, 10, 37, 36, 37, 10, 41, 24, 43, 25, 30, 44, 47, 7, 48, 34, 49, 31, 53, 23, 53, 14, 55, 56, 59, 17, 61, 60, 46, 7, 63, 40, 67, 43, 67, 40, 71, 11, 73, 72, 58, 49, 75, 48, 79, 11, 63, 80, 83, 25, 83, 84, 85, 26, 89, 33, 89, 61

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A033879, A187793, A296075, A318441, A318442, A318679.

f[n_] := 2 n - DivisorSigma[1, n]; Array[DivisorSum[#, f, f@ # >= 0 &] &, 92] (* Michael De Vlieger, Sep 04 2018 *)

A318678(n) = sumdiv(n,d,d=d+d-sigma(d); (d>0)*d);

a(n) = Sum_{d|n} [A033879(d) > 0]*A033879(d).

a(n) = A296075(n) + A318679(n).

A378531 Dirichlet convolution of A378432 and A378542.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 2, 2, 3, 0, 4, 0, 3, 3, 6, 0, 4, 0, 4, 3, 3, 0, 14, 2, 3, 2, 4, 0, 6, 0, 10, 3, 3, 3, 18, 0, 3, 3, 14, 0, 6, 0, 4, 4, 3, 0, 30, 2, 4, 3, 4, 0, 14, 3, 14, 3, 3, 0, 30, 0, 3, 4, 22, 3, 6, 0, 4, 3, 6, 0, 48, 0, 3, 4, 4, 3, 6, 0, 30, 6, 3, 0, 30, 3, 3, 3, 14, 0, 30, 3, 4, 3, 3, 3, 74, 0, 4, 4, 18

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Möbius transform of A378533.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A378532 (Dirichlet inverse), A378432, A378533 (inverse Möbius transform), A378542.

Cf. also A345182.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
memoA378432 = Map();
A378432(n) = if(1==n,1,my(v); if(mapisdefined(memoA378432,n,&v), v, v = -sumdiv(n,d,if(dA296075(n/d)*A378432(d),0)); mapput(memoA378432,n,v); (v)));
A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
A378531(n) = sumdiv(n,d,A378432(d)*A378542(n/d));

a(n) = Sum_{d|n} A378432(d)*A378542(n/d).

a(n) = Sum_{d|n} A008683(d)*A378533(n/d).

cp's OEIS Frontend

A378532 Dirichlet convolution of A296075 and A378525.

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A378432 Dirichlet inverse of A296075, where A296075 is the sum of deficiencies of divisors of n.

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A033879 Deficiency of n, or 2n - (sum of divisors of n).

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A066218 Numbers k such that sigma(k) = Sum_{j|k, j

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A296074 Sum of deficiencies of the proper divisors of n.

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A318679 Sum of abundancies of abundant divisors of n.

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A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

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