A326143 -id:A326143 - cp's OEIS frontend

A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - n).

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

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A000203, A007947, A066503, A326142, A326143, A326145.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140.

rad[n_] := Times @@ (First /@ FactorInteger@n);
a[n_] := GCD[n - rad[n], DivisorSigma[1, n] - rad[n] - n];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021 *)

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));

a(n) = gcd(A066503(n), A326143(n)) = gcd(n-A007947(n), A000203(n)-A007947(n)-n).

A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 234, 384, 496, 936, 1536, 1638, 6144, 8128, 24576, 42588, 98304, 393216, 1089270, 1572864, 6291456, 25165824, 33550336, 100663296, 115048440, 402653184, 1185125760, 1610612736

Offset: 1

Antti Karttunen, Jul 28 2020

Numbers k such that gcd(sigma(k)-A007947(k), A007947(k)) == A007947(k) are those in A175200. These are equal to k such that gcd(A326143(k), A007947(k)) = gcd(sigma(k)-A007947(k)-k, A007947(k)) are equal to A007947(k).
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

Intersection of A175200 and A336552.
Cf. A007947, A326143, A336551.
Cf. A000396, A002023, A326145 (subsequences).
Cf. also A336641 for a similar construction.

A007947(n) = factorback(factorint(n)[, 1]);
isA336550(n) = { my(r=A007947(n), s=sigma(n), u=((n/r)-1)); (!(s%r) && (gcd(u,(s-r-n))==u)); };

A336552 Numbers k such that A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero. Numbers k such that gcd(A336551(k), A326143(k)) is equal to A336551(k).

Original entry on oeis.org

4, 6, 12, 20, 24, 28, 44, 45, 48, 52, 60, 63, 68, 76, 84, 90, 92, 96, 99, 116, 117, 120, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 168, 171, 172, 188, 192, 198, 204, 207, 212, 220, 228, 234, 236, 244, 260, 261, 264, 268, 272, 276, 279, 284, 292, 294, 306, 308, 312, 315, 316, 325, 332, 333, 340, 342, 348, 350

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Numbers k such that either A336551(k) and A326143(k) are both zero (in which case k is squarefree), or A336551(k) divides A326143(k) (in which case k is not squarefree).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007947, A326143, A336550, A336551, A336553 (odd terms).

A007947(n) = factorback(factorint(n)[, 1]);
A326143(n) = (sigma(n)-A007947(n)-n);
A336551(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); (factorback(f)-1); };
isA336552(n) = { my(u=A336551(n)); (u==gcd(u,A326143(n))); };

A336553 Odd numbers k such that gcd(A336551(k), A326143(k)) is equal to A336551(k).

Original entry on oeis.org

45, 63, 99, 117, 147, 153, 171, 207, 261, 279, 315, 325, 333, 369, 387, 423, 425, 477, 495, 531, 549, 585, 603, 639, 657, 693, 711, 725, 735, 747, 765, 801, 819, 847, 855, 873, 909, 925, 927, 963, 981, 1017, 1025, 1035, 1071, 1125, 1143, 1179, 1197, 1233, 1251, 1287, 1305, 1325, 1341, 1359, 1395, 1413, 1449, 1467, 1503

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Is the intersection with A336554 empty?

Odd terms in A336552.

Cf. A336554.

A007947(n) = factorback(factorint(n)[, 1]);
A326143(n) = (sigma(n)-A007947(n)-n);
A336551(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); (factorback(f)-1); };
isA336553(n) = if(!(n%2),0, my(u=A336551(n)); (u==gcd(u,A326143(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

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

A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 1, 13, 10, 8, 1, 22, 1, 10, 9, 29, 1, 33, 1, 32, 11, 14, 1, 54, 26, 16, 37, 42, 1, 42, 1, 61, 15, 20, 13, 85, 1, 22, 17, 80, 1, 54, 1, 62, 63, 26, 1, 118, 50, 83, 21, 72, 1, 114, 17, 106, 23, 32, 1, 138, 1, 34, 83, 125, 19, 78, 1, 92, 27, 74, 1, 189, 1, 40, 109, 102, 19, 90, 1, 176, 118, 44, 1, 182, 23, 46, 33

Offset: 1

Antti Karttunen, Jun 09 2019

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, A007947, A066503, A326143.
Cf. also A326053, A326061, A326126.
Cf. A013661, A065463.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; a[n_] := DivisorSigma[1, n] - rad[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A326142(n) = (sigma(n)-A007947(n));

a(n) = A000203(n) - A007947(n).
a(n) = n + A326143(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065463 = 0.940491... . - Amiram Eldar, Dec 05 2023

A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

6, 28, 496, 936, 1638, 8128, 33550336

Offset: 1

Antti Karttunen, Jun 09 2019

Numbers n such that either A066503(n) and A326143(n) are both zero or A066503(n) is not zero and divides A326143(n).
Question: Are there any odd terms?
No other terms < 2^31.

Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396 (a subsequence), A007947, A066503, A228058, A326142, A326143, A326144.

A007947(n) = factorback(factorint(n)[, 1]);
isA326145(n) = { my(b=A007947(n), t=n-b, u = (sigma(n)-b)-n); (gcd(t,u)==t); };
\\ Or alternatively as:
isA326145(n) = { my(t=A326143(n), u=A066503(n)); ((!u && !t)||(u && !(t%u))); };

A336646 a(n) = n - A326144(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 39, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 61, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 73, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

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

Cf. A007947, A066503, A326143, A326144.
Cf. A326145 (positions where coincides with A007947).
Cf. A336555 (positions where differs from A336647).
Cf. also A336645, A336647.

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));
A336646(n) = (n - A326144(n));

cp's OEIS Frontend

A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - n).

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A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

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A336552 Numbers k such that A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero. Numbers k such that gcd(A336551(k), A326143(k)) is equal to A336551(k).

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A336553 Odd numbers k such that gcd(A336551(k), A326143(k)) is equal to A336551(k).

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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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A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

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A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

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