A323910 -id:A323910 - cp's OEIS frontend

A331180 Number of values of k, 1 <= k <= n, with A323910(k) = A323910(n), where A323910 is Dirichlet inverse of deficiency of n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 4, 5, 1, 1, 2, 1, 3, 1, 5, 1, 5, 1, 6, 1, 1, 1, 6, 1, 1, 2, 6, 1, 2, 1, 7, 1, 3, 1, 2, 1, 2, 8, 9, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 7, 1, 10, 1, 11, 2, 1, 2, 1, 1, 2, 2, 7, 1, 1, 1, 8, 2

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A323910.

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

Cf. A033879, A323910.

Cf. also A331178, A331181.

f[n_] := 2 n - DivisorSigma[1, n];
A323910[n_] := A323910[n] = If[n == 1, 1, -Sum[f[n/d] A323910[d], {d, Most@Divisors[n]}]];
Module[{b}, b[] = 0; a[n] := With[{t = A323910[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v331180 = ordinal_transform(DirInverse(vector(up_to,n,A033879(n))));
A331180(n) = v331180[n];

A378533 Dirichlet convolution of A323910 and A378542.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 5, 3, 4, 1, 10, 1, 4, 4, 11, 1, 10, 1, 10, 4, 4, 1, 26, 3, 4, 5, 10, 1, 16, 1, 21, 4, 4, 4, 34, 1, 4, 4, 26, 1, 16, 1, 10, 10, 4, 1, 62, 3, 10, 4, 10, 1, 26, 4, 26, 4, 4, 1, 56, 1, 4, 10, 43, 4, 16, 1, 10, 4, 16, 1, 98, 1, 4, 10, 10, 4, 16, 1, 62, 11, 4, 1, 56, 4, 4, 4, 26, 1, 56, 4, 10, 4, 4, 4

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Inverse Möbius transform of A378531.

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

Cf. A323910, A378542.
Cf. A378531 (Möbius transform), A378534 (Dirichlet inverse).
Cf. also A378223.

A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
A033879(n) = (n+n-sigma(n));
memoA323910 = Map();
A323910(n) = if(1==n,1,my(v); if(mapisdefined(memoA323910,n,&v), v, v = -sumdiv(n,d,if(dA033879(n/d)*A323910(d),0)); mapput(memoA323910,n,v); (v)));
A378533(n) = sumdiv(n,d,A323910(d)*A378542(n/d));

a(n) = Sum_{d|n} A323910(d)*A378542(n/d).

a(n) = Sum_{d|n} A378531(d).

A378754 Dirichlet convolution of -A252748 and A323910.

Original entry on oeis.org

1, 0, -1, -2, -1, -2, -3, -10, -10, -2, -1, -10, -3, -6, -5, -38, -1, -22, -3, -10, -11, -2, -5, -42, -14, -6, -60, -22, -1, -14, -5, -130, -5, -2, -11, -76, -3, -6, -11, -42, -1, -30, -3, -10, -36, -10, -5, -158, -46, -30, -5, -22, -5, -144, -5, -78, -11, -2, -1, -58, -5, -10, -64, -422, -11, -14, -3, -10, -17, -30

Offset: 1

Antti Karttunen, Dec 11 2024

sign

Cf. A033879, A252748, A323910, A378755 (Dirichlet inverse).

A033879(n) = (n+n-sigma(n));
memoA323910 = Map();
A323910(n) = if(1==n,1,my(v); if(mapisdefined(memoA323910,n,&v), v, v = -sumdiv(n,d,if(dA033879(n/d)*A323910(d),0)); mapput(memoA323910,n,v); (v)));
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A252748(n) = (A003961(n) - (2*n));
A378754(n) = sumdiv(n,d,-A252748(d)*A323910(n/d));

a(n) = Sum_{d|n} -A252748(d)*A323910(n/d).

A379106 Dirichlet convolution of A000120 and A323910, where A323910 is the Dirichlet inverse of the deficiency of n, and A000120 is the binary weight of n.

Original entry on oeis.org

1, 0, 0, 0, -2, 2, -3, 0, -3, 2, -7, 4, -9, 2, 2, 0, -14, 4, -15, 4, -1, 2, -18, 8, -8, 2, -4, 4, -24, 2, -25, 0, -2, 2, 5, 12, -33, 2, 0, 8, -37, 0, -38, 4, 10, 2, -41, 16, -20, 0, 2, 4, -48, 2, 15, 8, 0, 2, -53, 12, -55, 2, 19, 0, 16, -8, -63, 4, -3, -4, -66, 32, -69, 2, -2, 4, 18, -12, -73, 16, -15, 2, -78, 8, 30

Offset: 1

Antti Karttunen, Dec 16 2024

sign

Cf. A000120, A033879, A323910, A379107 (Dirichlet inverse).

Cf. also A294898, A378754, A378756.

A033879(n) = (n+n-sigma(n));
memoA323910 = Map();
A323910(n) = if(1==n,1,my(v); if(mapisdefined(memoA323910,n,&v), v, v = -sumdiv(n,d,if(dA033879(n/d)*A323910(d),0)); mapput(memoA323910,n,v); (v)));
A379106(n) = sumdiv(n,d,hammingweight(d)*A323910(n/d));

a(n) = Sum_{d|n} A000120(d)*A323910(n/d).

A378654 Abundant numbers k for which A323910(k) = A378643(k).

Original entry on oeis.org

162, 460, 784, 972, 1000, 1584, 1760, 1944, 4136, 4928, 6860, 11128, 30976

Offset: 1

Antti Karttunen, Dec 03 2024

Cf. A103977, A323910, A378643.

Subsequence of A005101.

is_A378654(n) = ((sigma(n)>2*n) && (A323910(n)==A378643(n)));

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

A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

Original entry on oeis.org

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55

Offset: 1

Alonso del Arte, May 26 2011

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A191161, Sequence Machine.

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).

Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)

a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011
From Antti Karttunen, Nov 22 2024: (Start)
Following formulas were conjectured by Sequence Machine:
For n > 1, a(n) = A191150(n) + A074206(n).
a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.
a(n) = Sum_{d|n} A330575(d).
a(n) = Sum_{d|n} d*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A074206(n/d).
a(n) = Sum_{d|n} A007429(d)*A174725(n/d).
a(n) = Sum_{d|n} A000010(d)*A253249(n/d).
a(n) = Sum_{d|n} A038040(d)*A323912(n/d).
a(n) = Sum_{d|n} A060640(d)*A323910(n/d).
(End)

A346246 Dirichlet inverse of A344587, 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, -2, -4, -1, -6, 10, -10, -2, -3, 14, -12, 4, -16, 22, 26, -4, -18, 2, -22, 6, 42, 26, -28, 6, -5, 34, -6, 10, -30, -66, -36, -8, 50, 38, 62, 7, -40, 46, 66, 10, -42, -106, -46, 12, 14, 58, -52, 8, -9, 2, 74, 16, -58, -2, 74, 18, 90, 62, -60, -18, -66, 74, 26, -16, 98, -126, -70, 18, 114, -150, -72, 18, -78, 82, 12, 22

Offset: 1

Antti Karttunen, Jul 19 2021

Dirichlet inverse of the deficiency of prime shifted n.

Cf. A000203, A003961, A003973, A323910, A344587, A346247, A346251 (positions of zeros).

Cf. also A346235, A346248, A346254.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };
v346246 = DirInverseCorrect(vector(up_to,n,A344587(n)));
A346246(n) = v346246[n];

a(n) = A323910(A003961(n)).

a(n) = A346247(n) - A344587(n).

A346248 Dirichlet inverse of -A252748, 2*n - A003961(n).

Original entry on oeis.org

1, -1, -1, 2, -3, 5, -3, 8, 8, 7, -9, 10, -9, 11, 11, 32, -15, 16, -15, 6, 19, 13, -17, 48, 8, 17, 56, 18, -27, -3, -25, 128, 17, 19, 25, 104, -33, 23, 25, 32, -39, 9, -39, -6, 24, 29, -41, 224, 32, 16, 23, 6, -47, 144, 35, 88, 31, 31, -57, 78, -55, 37, 72, 512, 43, -33, -63, -18, 41, -13, -69, 512, -67, 41, 40, -6, 43

Offset: 1

Antti Karttunen, Jul 19 2021

Zeros occur at n = 352, 26840, 34816, 3787168, ...

Antti Karttunen, Table of n, a(n) for n = 1..34816
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A252748, A346250.

Cf. also A323910, A346235, A346246, A346254.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
v346248 = DirInverseCorrect(vector(up_to,n,(n+n)-A003961(n)));
A346248(n) = v346248[n];

a(n) = A346250(n) + A252748(n).

A323911 Sum of deficiency of n (A033879) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 8, 0, -2, 0, 12, 16, 1, 0, -2, 0, 0, 24, 20, 0, -8, 16, 24, 12, 2, 0, -28, 0, 1, 40, 32, 48, -15, 0, 36, 48, -6, 0, -36, 0, 6, 16, 44, 0, -20, 36, 6, 64, 8, 0, -12, 80, -4, 72, 56, 0, -46, 0, 60, 28, 1, 96, -52, 0, 12, 88, -44, 0, -39, 0, 72, 28, 14, 120, -60, 0, -18, 37, 80, 0, -58, 128, 84, 112, 0, 0, -52, 144, 18

Offset: 1

Antti Karttunen, Feb 12 2019

sign

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

Cf. A033879, A297159, A323403, A323910, A323913.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];
A323911(n) = (A033879(n)+A323910(n));

a(n) = A033879(n) + A323910(n).

cp's OEIS Frontend

A331180 Number of values of k, 1 <= k <= n, with A323910(k) = A323910(n), where A323910 is Dirichlet inverse of deficiency of n.

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A378533 Dirichlet convolution of A323910 and A378542.

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A378754 Dirichlet convolution of -A252748 and A323910.

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A379106 Dirichlet convolution of A000120 and A323910, where A323910 is the Dirichlet inverse of the deficiency of n, and A000120 is the binary weight of n.

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A378654 Abundant numbers k for which A323910(k) = A378643(k).

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

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A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

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A346246 Dirichlet inverse of A344587, 2*A003961(n) - sigma(A003961(n)).

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