A325814 -id:A325814 - cp's OEIS frontend

A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).

1, 2, 3, 1, 5, 6, 7, -1, 3, 10, 11, 0, 13, 14, 15, -5, 17, 0, 19, 2, 21, 22, 23, -12, 10, 26, 3, 4, 29, 30, 31, -13, 33, 34, 35, -24, 37, 38, 39, -14, 41, 42, 43, 8, 9, 46, 47, -36, 21, 5, 51, 10, 53, -18, 55, -16, 57, 58, 59, -12, 61, 62, 15, -29, 65, 66, 67, 14, 69, 70, 71, -72, 73, 74, 15, 16, 77, 78, 79, -46, 3, 82, 83, -12, 85

Offset: 1

Antti Karttunen, Jun 02 2019

sign

Question: Are a(12) = 0 and a(18) = 0 the only zeros in this sequence?

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

Cf. A000203, A033879, A034448, A048146, A162296, A325314, A325814, A325973, A325974, A325975, A325977, A325979, A325981.

Cf. A002117, A013661.

Table[(1/2) If[n == 1, 2, 2 n - DivisorSigma[1, n] + Times @@ (1 + FactorInteger[n][[;; , 1]]) - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &]], {n, 85}] (* Michael De Vlieger, Jun 06 2019 *)

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);

a(n) = (1/2)*(A325314(n) + A325814(n)).
a(n) = n - A325974(n).
a(n) = A033879(n) + A325977(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/4 - zeta(2)*(1/2 - 1/(4*zeta(3))) = 0.2696411609... . - Amiram Eldar, Feb 22 2024

A325824 a(n) = A325814(A228058(n)).

Original entry on oeis.org

27, 75, 99, 203, 171, 255, 219, 243, 171, 335, 315, 363, 539, 539, 435, 575, 531, 707, 767, 579, 603, 735, 651, 675, 815, 507, 819, 1055, 891, 675, 939, 1211, 1343, 1215, 1035, 1419, 1083, 1155, 1179, 1691, 1547, 1455, 1715, 1859, 1371, 1395, 1443, 759, 1775, 1539, 1179, 1611, 1935, 1659, 2015, 1683, 2219, 1755, 2507, 2175, 1875, 2255

Offset: 1

Antti Karttunen, May 23 2019

sign

First negative term occurs as a(16307) = -210973, with A228058(16307) = 1289925. The next negative terms occurs as a(20807) = -242901, with A228058(20807) = 1686825.

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A228058, A325379, A325814, A325822, A325823.

Cf. also A325320.

up_to = 10000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, May 23 2019
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325824(n) = A325814(A228058(n));

a(n) = A325814(A228058(n)).

a(n) = A325379(n) + A325823(n).

A325822 Terms k of A228058 for which A325814(k) is a multiple of A034460(k).

Original entry on oeis.org

477, 3725, 29161, 107797, 166753, 205409, 500837, 535277, 780625, 1610389, 5649841, 6968125, 10292809, 10633429, 24231241, 32771201, 38322857, 40028661, 104861501, 170384117, 183593125, 277405641, 326081953, 488265625, 491716541, 704531953, 797338489, 836737393, 2053219321, 2359421369, 3012238153

Offset: 1

Antti Karttunen, May 23 2019

nonn

Such terms A228058(n) that A325823(n) is a divisor of A325824(n).
If any odd perfect number exists, then it must occur in this sequence.
This is not a subsequence of A325376: 107797 is the first term that does not occur there.

Index entries for sequences where any odd perfect numbers must occur

Cf. A034460, A228058, A325376, A325812, A325814, A325823, A325824.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
for(n=1,oo, if(isA228058(n) && !(A325814(n)%A034460(n)), print1(n, ", ")));

A325810 Lexicographically earliest sequence such that a(i) = a(j) => A034460(i) = A034460(j) and A325814(i) = A325814(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 35, 2, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 45, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Antti Karttunen, May 23 2019

nonn

Restricted growth sequence transform of the ordered pair [A034460(n), A325814(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A033879(i) = A033879(j),
  a(i) = a(j) => A325811(i) = A325811(j) => A325813(i) = A325813(i).

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

Cf. A033879, A034460, A324400, A325811, A325813, A325814.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
v325810 = rgs_transform(vector(up_to,n,[A034460(n), A325814(n)]));
A325810(n) = v325810[n];

A325811 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) where f(n) = [A034460(n), A325814(n)] for all other numbers, except f(p) = -1 for odd primes.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 5, 6, 3, 7, 3, 8, 9, 2, 3, 10, 3, 11, 12, 13, 3, 14, 15, 16, 17, 18, 3, 19, 3, 2, 20, 21, 22, 23, 3, 24, 25, 26, 3, 27, 3, 28, 29, 30, 3, 31, 32, 33, 34, 35, 3, 36, 37, 38, 39, 40, 3, 41, 3, 42, 25, 2, 43, 44, 3, 45, 46, 47, 3, 48, 3, 49, 50, 51, 52, 53, 3, 54, 32, 55, 3, 56, 57, 58, 59, 60, 3, 61, 62, 63, 64, 65, 66, 67, 3, 68, 69, 70, 3

Offset: 1

Antti Karttunen, May 23 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  A325810(i) = A325810(j) => a(i) = a(j) => A325813(i) = A325813(i).

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

Cf. A034460, A305801, A325810, A325813, A325814.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
Aux325811(n) = if(isprime(n)&&(n%2),-(n%2),[A034460(n), A325814(n)]);
v325811 = rgs_transform(vector(up_to,n,Aux325811(n)));
A325811(n) = v325811[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

A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

Original entry on oeis.org

45, 117, 153, 245, 261, 325, 333, 369, 405, 425, 477, 549, 605, 637, 657, 725, 801, 833, 845, 873, 909, 925, 981, 1017, 1025, 1053, 1233, 1325, 1341, 1377, 1413, 1421, 1445, 1525, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2057, 2061, 2097, 2169

Offset: 1

T. D. Noe, Aug 13 2013

It has been proved that if an odd perfect number exists, it belongs to this sequence. The first term of the form p^5 * n^2 is 28125 = 5^5 * 3^2, occurring in position 520.
Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - T. D. Noe, Aug 15 2013
Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - Antti Karttunen, Jun 13 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
For various subsequences with additional conditions, see A228059, A325376, A325380, A325822, A326137 (with omega(n)>=5), A324898 (conjectured, subsequence if it does not contain any prime powers), A354362, A386425 (conjectured), A386427 (nondeficient terms), A386428 (powerful terms), A386429 U A351574.
Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

import Data.List (partition)
a228058 n = a228058_list !! (n-1)
a228058_list = filter f [1, 3 ..] where
   f x = length us == 1 && not (null vs) &&
         fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
         where (us,vs) = partition (odd . snd) $
                         zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Aug 14 2013

nn = 100; n = 1; t = {}; While[Length[t] < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1,1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *)

up_to = 1000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

From Antti Karttunen, Apr 22 2019 & Jun 03 2019: (Start)
A325313(a(n)) = -A325319(n).
A325314(a(n)) = -A325320(n).
A001065(a(n)) = A325377(n).
A033879(a(n)) = A325379(n).
A034460(a(n)) = A325823(n).
A325814(a(n)) = A325824(n).
A324213(a(n)) = A325819(n).
(End)

Note in parentheses added to the definition by Antti Karttunen, Jun 03 2019

A325975 a(n) = gcd(A325977(n), A325978(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

See comments in A325979 and A325981.

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

Cf. A034460, A048107, A325313, A325314, A325814, A325973, A325974, A325977, A325978, A325979, A325981.

Cf. also A009194, A325385, A325813, A326046, A326047, A326048, A326056, A326057, A326060, A326062.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325313(n) = (A048250(n) - n);
A325314(n) = (n - A162296(n));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325977(n) = ((A034460(n)+A325313(n))/2);
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
\\ Or alternatively, as:
A325975(n) = (1/2)*gcd((A034460(n)+A325313(n)),(A325814(n)+A325314(n)));

a(n) = gcd(A325977(n), A325978(n)).

a(n) = (1/2)*gcd(A034460(n)+A325313(n), A325814(n)+A325314(n)).

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 12, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 7, 3, 6, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 12, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 21, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 23 2019

nonn

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

Cf. A034460, A323166, A325812, A325814.

Cf. also A325385.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));

a(n) = gcd(A034460(n), A325814(n)).

A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

Original entry on oeis.org

45, 495, 585, 765, 855, 1305, 18837, 21525, 31635, 38295, 45315, 50445, 51255, 60435, 63495, 68085, 77265, 96615, 1403115, 2446353, 3411975, 3999465, 4091745, 4233537, 4287255, 4631319, 10813425, 10967085, 11490345, 15578199, 16143309, 16329645, 16633071, 17179515, 17311203, 17355915, 21159075, 21933975, 22579725

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977 and A325978 are never zero on same n, these are odd composite numbers n such that A325977(n) is not zero and divides A325978(n).
Based on the first 147 terms it seems that this sequence is a subsequence of A072357, that is each term has exactly one prime factor with exponent 2, with one or more other prime factors that are all unitary (i.e., each term satisfies A001222(x) - A001221(x) = 1). On the other hand, it has been proved that no odd perfect number, if such numbers exist at all, can have such a factorization (see A326137 and a link to P. P. Nielsen's paper there).
Nineteen initial terms factorize as:
= 3^2 * 5^1,
= 3^2 * 5^1 * 11^1,
= 3^2 * 5^1 * 13^1,
= 3^2 * 5^1 * 17^1,
= 3^2 * 5^1 * 19^1,
= 3^2 * 5^1 * 29^1,
= 3^2 * 7^1 * 13^1 * 23^1,
= 3^1 * 5^2 * 7^1 * 41^1,
= 3^2 * 5^1 * 19^1 * 37^1,
= 3^2 * 5^1 * 23^1 * 37^1,
= 3^2 * 5^1 * 19^1 * 53^1,
= 3^2 * 5^1 * 19^1 * 59^1,
= 3^2 * 5^1 * 17^1 * 67^1,
= 3^2 * 5^1 * 17^1 * 79^1,
= 3^2 * 5^1 * 17^1 * 83^1,
= 3^2 * 5^1 * 17^1 * 89^1,
= 3^2 * 5^1 * 17^1 * 101^1,
= 3^2 * 5^1 * 19^1 * 113^1,
  1403115 = 3^1 * 5^1 * 7^2 * 23^1 * 83^1,
and the 62nd term as a(62) = 2919199437 = 3^2 * 7^1 * 11^1 * 43^1 * 163^1 * 601^1.
If we select a subsequence of terms for which the quotient A325978(n)/A325977(n) is positive, then we are left with the following numbers: 495, 585, 31635, 38295, 45315, 51255, 60435, 63495, 1403115, 3999465, etc. which is a subsequence of A326070.

Antti Karttunen (terms 1-62) & Giovanni Resta, Table of n, a(n) for n = 1..147
Index entries for sequences where any odd perfect numbers must occur

Cf. A072357, A228058, A325973, A325974, A325975, A325977, A325978, A325979, A326064, A326070, A326071, A326137.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
isA325981(n) = ((n%2)&&!isprime(n)&&(A325975(n)==abs(A325977(n))));
\\ Or alternatively as:
isA325981(n) = if(!(n%2)||isprime(n),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(x&&!(y%x)));

cp's OEIS Frontend

A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).

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A325824 a(n) = A325814(A228058(n)).

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A325822 Terms k of A228058 for which A325814(k) is a multiple of A034460(k).

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A325810 Lexicographically earliest sequence such that a(i) = a(j) => A034460(i) = A034460(j) and A325814(i) = A325814(j) for all i, j.

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A325811 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) where f(n) = [A034460(n), A325814(n)] for all other numbers, except f(p) = -1 for odd primes.

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

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A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

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A325975 a(n) = gcd(A325977(n), A325978(n)).