A326128 -id:A326128 - cp's OEIS frontend

A326129 a(n) = gcd(A326127(n), A326128(n)).

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 1, 12, 4, 6, 1, 16, 1, 18, 1, 10, 8, 22, 6, 1, 10, 2, 21, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 1, 4, 20, 46, 1, 1, 1, 30, 3, 52, 12, 38, 2, 34, 26, 58, 3, 60, 28, 2, 1, 46, 12, 66, 1, 42, 4, 70, 1, 72, 34, 2, 3, 58, 12, 78, 1, 1, 38, 82, 7, 62, 40, 54, 2, 88, 2, 70, 1, 58, 44, 70, 30

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Question: Are there any other numbers than those in A000396 that satisfy a(k) = A326128(k)?

See also comments in A336641, where all such k should reside. - Antti Karttunen, Jul 29 2020

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

Cf. A000396, A007913, A326126, A326127, A326128.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326130, A326140, A326144, A336641, A336645.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A326129(n) = gcd(A326127(n), A326128(n));

a(n) = n - A336645(n). - Antti Karttunen, Jul 29 2020

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

A326127 a(n) = A326126(n) - n, where A326126 gives the sum of all other divisors of n except the squarefree part of n.

Original entry on oeis.org

-1, -1, -2, 2, -4, 0, -6, 5, 3, -2, -10, 13, -12, -4, -6, 14, -16, 19, -18, 17, -10, -8, -22, 30, 5, -10, 10, 21, -28, 12, -30, 29, -18, -14, -22, 54, -36, -16, -22, 40, -40, 12, -42, 29, 28, -20, -46, 73, 7, 41, -30, 33, -52, 60, -38, 50, -34, -26, -58, 93, -60, -28, 34, 62, -46, 12, -66, 41, -42, 4, -70, 121, -72, -34, 46, 45, -58, 12

Offset: 1

Antti Karttunen, Jun 09 2019

sign

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

Cf. A000203, A001065, A007913, A033879, A326126, A326128, A326129.

Cf. also A326143.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^Mod[e, 2]; a[n_] := Module[{f = FactorInteger[n]}, Times @@ f1 @@@ f - Times @@ f2 @@@ f - n]; a[1] = -1; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326127(n) = (sigma(n)-core(n)-n);
```

a(n) = A000203(n) - A007913(n) - n = A001065(n) - A007913(n).
a(n) = A326128(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 - 1/2 = -0.00651977... . - Amiram Eldar, Mar 21 2024

A326126 Sum of all other divisors of n except the squarefree part of n: a(n) = sigma(n) - A007913(n).

Original entry on oeis.org

0, 1, 1, 6, 1, 6, 1, 13, 12, 8, 1, 25, 1, 10, 9, 30, 1, 37, 1, 37, 11, 14, 1, 54, 30, 16, 37, 49, 1, 42, 1, 61, 15, 20, 13, 90, 1, 22, 17, 80, 1, 54, 1, 73, 73, 26, 1, 121, 56, 91, 21, 85, 1, 114, 17, 106, 23, 32, 1, 153, 1, 34, 97, 126, 19, 78, 1, 109, 27, 74, 1, 193, 1, 40, 121, 121, 19, 90, 1, 181, 120, 44, 1, 203, 23, 46, 33, 158, 1, 224, 21

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537.
Index entries for sequences related to sigma(n).

Cf. A000203, A007913, A326127, A326128, A326129.

Cf. also A326053, A326061, A326142.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^Mod[e, 2]; a[n_] := Module[{f = FactorInteger[n]}, Times @@ f1 @@@ f - Times @@ f2 @@@ f]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326126(n) = (sigma(n)-core(n));
```

a(n) = A000203(n) - A007913(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = 0.4934802... . - Amiram Eldar, Mar 21 2024

A336645 a(n) = n - A326129(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 11, 1, 10, 9, 15, 1, 17, 1, 19, 11, 14, 1, 18, 24, 16, 25, 7, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 43, 41, 26, 1, 47, 48, 49, 21, 49, 1, 42, 17, 54, 23, 32, 1, 57, 1, 34, 61, 63, 19, 54, 1, 67, 27, 66, 1, 71, 1, 40, 73, 73, 19, 66, 1, 79, 80, 44, 1, 77, 23, 46, 33, 86, 1, 88, 21

Offset: 1

Antti Karttunen, Jul 29 2020

nonn

It seems that A000396 gives all such n for which a(n) = A007913(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A000396, A007913, A326126, A326127, A326128, A326129.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A336645(n) = (n-gcd(A326127(n), A326128(n)));

a(n) = n - A326129(n) = n - gcd(A326127(n), A326128(n)).

A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 150, 294, 384, 496, 1014, 1536, 3276, 3750, 3780, 6144, 8128, 14406, 20328, 24576, 32760, 93750, 98304, 171366, 306180, 393216, 705894, 1241460, 1572864, 2343750, 6291456, 16380000, 24800580, 25165824, 28960854, 30387840, 33550336, 34588806, 58593750, 100663296, 165143160, 332226048, 402653184

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Numbers k such that A326128(k) = A326129(k) form a subsequence of this sequence. So far it is not known whether it contains any other terms apart from those of A000396. See comments in A326129.
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A007913, A228058, A326126, A326127, A326128, A326129, A336642.
Cf. A000396, A002023 (subsequences).
Cf. also A336550 for a similar construction.

isA336641(n) = { my(c=core(n), s=sigma(n), u=((n/c)-1)); (!(s%c) && (gcd(u,(s-c-n))==u)); };

cp's OEIS Frontend

A326129 a(n) = gcd(A326127(n), A326128(n)).

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

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A326127 a(n) = A326126(n) - n, where A326126 gives the sum of all other divisors of n except the squarefree part of n.

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A326126 Sum of all other divisors of n except the squarefree part of n: a(n) = sigma(n) - A007913(n).

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A336645 a(n) = n - A326129(n).

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A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

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