A318879 -id:A318879 - cp's OEIS frontend

A326140 a(n) = gcd(A318878(n), A318879(n)).

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 2, 12, 2, 6, 1, 16, 1, 18, 2, 10, 2, 22, 2, 19, 2, 14, 6, 28, 6, 30, 1, 18, 2, 22, 1, 36, 2, 22, 2, 40, 2, 42, 2, 12, 2, 46, 2, 41, 1, 30, 6, 52, 2, 38, 2, 34, 2, 58, 6, 60, 2, 22, 1, 46, 6, 66, 2, 42, 2, 70, 1, 72, 2, 26, 6, 58, 2, 78, 2, 41, 2, 82, 2, 62, 2, 54, 2, 88, 6, 70, 2, 58, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A033879, A083254, A318878, A318879, A326141.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326144.

A326140(n) = { my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); gcd(t,u); };

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);
A318879(n) = sumdiv(n,d,d=d-(2*eulerphi(d)); (d>0)*d);
A326140(n) = gcd(A318878(n), A318879(n));

A326141 Odd numbers n for which A318879(n) is not zero and A318879(n) divides A318878(n); odd numbers such that A326140(n) = A318879(n).

Original entry on oeis.org

105, 195, 4785, 22515, 56865, 228285, 237315, 484245, 671853, 1838145, 1946955, 3446895, 4522695, 12955245, 37730865, 52475055, 53568885, 87612975

Offset: 1

Antti Karttunen, Jun 09 2019

Not a subsequence of A036798, even though many terms are members.

Questions: Are all terms multiples of three? Multiples of 3^(2k+1) but not of 3^(2k)? Are any of the terms included in A228058, A326137?

Index entries for sequences where any odd perfect numbers must occur

Cf. A033879, A083254, A036798, A228058, A318878, A318879, A326137, A326140.

Cf. also A325981, A326131.

isA326141(n) = if(!(n%2),0, my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); (gcd(t,u)==u));

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

A318878 Sum of A083254(d) for all such divisors d of n for which A083254(d) > 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 4, 10, 2, 12, 6, 6, 1, 16, 5, 18, 4, 10, 10, 22, 2, 19, 12, 14, 6, 28, 6, 30, 1, 18, 16, 22, 5, 36, 18, 22, 4, 40, 10, 42, 10, 12, 22, 46, 2, 41, 19, 30, 12, 52, 14, 38, 6, 34, 28, 58, 6, 60, 30, 22, 1, 46, 18, 66, 16, 42, 22, 70, 5, 72, 36, 26, 18, 58, 22, 78, 4, 41, 40, 82, 10, 62, 42, 54, 10, 88, 12, 70, 22, 58, 46, 70, 2

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

			n = 105 has divisors [1, 3, 5, 7, 15, 21, 35, 105]. When A083254 is applied to them, we obtain [1, 1, 3, 5, 1, 3, 13, -9]. Summing the positive numbers present, we get a(105) = 1+1+3+5+1+3+13 = 27.

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

Cf. A000010, A033879, A083254, A318879.

Cf. also A318678, A318874, A318876.

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);

a(n) = Sum_{d|n} [A083254(d) > 0]*A083254(d), where A083254(n) = 2*phi(n) - n, and [ ] are the Iverson brackets.

a(n) = A318879(n) + A033879(n).

A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, 0, 14, 0, 0, 5, 0, 0, 1, 0, 0, 0, 13, 5, 0, 0, 0, 1, 21, 0, 14, 0, 0, 0, 0, 0, 6, 8, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 5, 0, 12, 14, 0, 0, 17, 0, 0, 26, 74, 0, 350, 40, 0, 14, 53, 0, 0, 0, 70, 0, 0, 13, 18, 7, 0, 0, 0, 0, 15

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A323244, A329638, A329640, A329641, A329644.

Cf. also A318879.

A329639(n) = sumdiv(n,d,if((d=A329644(d))<0,-d,0));

a(n) = -Sum_{d|n} [A329644(d) < 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A329638(n) - A323244(n).

A318875 Number of divisors d of n for which 2*phi(d) < d.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 0, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 2, 0, 1, 0, 4, 0, 2, 0, 2, 0, 3, 0, 3, 0, 1, 0, 6, 0, 1, 0, 0, 0, 3, 0, 2, 0, 3, 0, 6, 0, 1, 0, 2, 0, 3, 0, 4, 0, 1, 0, 6, 0, 1, 0, 3, 0, 5, 0, 2, 0, 1, 0, 5, 0, 2, 0, 4, 0, 3, 0, 3, 1

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

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

Cf. A000010, A083254, A318874, A318877, A318879.

A318875 := n -> nops(select(d -> (2*numtheory:-phi(d)) < d, divisors(n))):
seq(A318875(n), n=1..199); # Peter Luschny, Sep 05 2018

A318875[n_] := DivisorSum[n, 1 &, 2*EulerPhi[#] < # &];
Array[A318875, 100] (* Paolo Xausa, Jul 08 2024 *)

A318875(n) = sumdiv(n,d,(2*eulerphi(d))

a(n) = Sum_{d|n} [A083254(d) < 0].

For all n >= 1, a(n) + A318874(n) + A007814(n) = A000005(n).

A318877 Sum of divisors d of n for which 2*phi(d) < d.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 10, 0, 18, 0, 14, 0, 0, 0, 24, 0, 30, 0, 22, 0, 42, 0, 26, 0, 42, 0, 46, 0, 0, 0, 34, 0, 72, 0, 38, 0, 70, 0, 62, 0, 66, 0, 46, 0, 90, 0, 60, 0, 78, 0, 78, 0, 98, 0, 58, 0, 138, 0, 62, 0, 0, 0, 94, 0, 102, 0, 94, 0, 168, 0, 74, 0, 114, 0, 110, 0, 150, 0, 82, 0, 186, 0, 86, 0, 154, 0, 154, 0, 138, 0, 94, 0

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

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

Cf. A000010, A000203, A006519, A083254, A318875, A318876, A318879.

Cf. also A187795.

A318877[n_] := DivisorSum[n,  # &, 2*EulerPhi[#] < # &];
Array[A318877, 100] (* Paolo Xausa, Jul 08 2024 *)

A318877(n) = sumdiv(n,d,((2*eulerphi(d))

a(n) = Sum_{d|n} [2*phi(d) < d]*d, where [ ] are the Iverson brackets.

For all n >= 1, a(n) + A318876(n) + 2*(A006519(n)-1) = A000203(n).

cp's OEIS Frontend

A326140 a(n) = gcd(A318878(n), A318879(n)).

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A326141 Odd numbers n for which A318879(n) is not zero and A318879(n) divides A318878(n); odd numbers such that A326140(n) = A318879(n).

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

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A318878 Sum of A083254(d) for all such divisors d of n for which A083254(d) > 0.

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A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

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A318875 Number of divisors d of n for which 2*phi(d) < d.

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A318877 Sum of divisors d of n for which 2*phi(d) < d.

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