A325379 -id:A325379 - cp's OEIS frontend

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

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

A228059 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 that are closer to being perfect than previous terms.

Original entry on oeis.org

45, 405, 2205, 26325, 236925, 1380825, 1660725, 35698725, 3138290325, 29891138805, 73846750725, 194401220013, 194509436121, 194581580193, 194689796301, 194798012409, 194906228517, 194942300553, 195230876841, 195339092949, 195447309057, 195699813309

Offset: 1

T. D. Noe, Aug 14 2013

nonn

A number x is perfect if sigma(x) = 2x, where sigma is the sum of divisors of x. See A228058 for numbers of the form p^(1+4k) * r^2. This sequence ends when the first odd perfect number occurs.
The first two papers by Dris listed below are for information only; this sequence in independent of the papers. In the second paper, Dris attempts to prove that the exponent of p above is 1 for odd perfect numbers. Coincidently, the first 9 numbers in this sequence have exponent 1.
a(38) > 10^12. - Giovanni Resta, Aug 16 2018
a(38) <= 283665529390725 = 15349 * (3^3 * 5 * 19 * 53)^2. - Giovanni Resta, Aug 23 2018
a(39) <= 3116918388785625 = 37993 * (3^2 * 5^2 * 19 * 67)^2. - Alexander Violette, Mar 05 2022
The first 37 terms are all multiples of 3, as well as the two additional terms given above. See also comments in A349752. - Antti Karttunen, Jan 04 2025

			           45 =   5 * 3^2.
          405 =   5 * 3^4.
         2205 =   5 * (3 * 7)^2.
        26325 =  13 * (3^2 * 5)^2.
       236925 =  13 * (3^3 * 5)^2.
      1380825 =  17 * (3 * 5 * 19)^2.
      1660725 =  61 * (3 * 5 * 11)^2.
     35698725 =  61 * (3^2 * 5 * 17)^2.
   3138290325 =  53 * (3^4 * 5 * 19)^2.
  29891138805 =   5 * (3^2 * 11^2 * 71)^2.
  73846750725 = 509 * (3 * 5 * 11 * 73)^2.

Giovanni Resta, Table of n, a(n) for n = 1..37
Jose Arnaldo B. Dris, The abundancy index of divisors of odd perfect numbers, J. Integer Sequences, 15 (2012), Article 12.4.4.
Jose Arnaldo B. Dris, A short "proof" for Sorli's conjecture on odd perfect numbers, arxiv 1308.2156 [math.NT], 2013-2015.
Jose Arnaldo B. Dris, Euclid-Euler Heuristics for (Odd) Perfect Numbers, arXiv preprint arXiv:1310.5616 [math.NT], 2013-2017.
Jose Arnaldo B. Dris, A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1311.6803 [math.NT], 2013-2015.
Jose Arnaldo Bebita Dris, 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 where any odd perfect numbers must occur

Cf. A000203 (sigma), A000396 (perfect numbers), A228058, A325379, A349752.

Cf. also A171929.

nn = 7; f[n_] := Abs[DivisorSigma[1, n]/n - 2]; n = 45; t = {n}; lastF = f[n]; cnt = 1; While[cnt < 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 && f[n] < lastF, cnt++; lastF = f[n]; Print[{n, lastF}]; AppendTo[t, n]]]; t

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));
m=-1; n=0; while(m!=0, n++; if(isA228058(n), if((m<0) || abs((sigma(n)/n)-2)Antti Karttunen, Apr 22 2019

a(10) (as communicated by T. D. Noe) added by Jose Arnaldo Bebita Dris, Aug 16 2018

a(11)-a(22) from Giovanni Resta, Aug 16 2018

A325320 Sum of proper divisors of A228058(n) that are not squarefree; a(n) = -A325314(A228058(n)).

Original entry on oeis.org

9, 9, 9, 49, 9, 25, 9, 9, 297, 25, 9, 9, 121, 49, 9, 25, 9, 49, 169, 9, 9, 25, 9, 9, 25, 585, 9, 25, 9, 729, 9, 49, 289, 25, 9, 121, 9, 9, 9, 361, 49, 25, 49, 121, 9, 9, 9, 2049, 25, 9, 1161, 9, 25, 9, 25, 9, 49, 9, 529, 25, 9, 25, 9, 169, 2381, 49, 1449, 9, 9, 9, 1593, 9, 25, 9, 121, 9, 49, 9, 2889, 9, 25, 289, 9, 2997, 9

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

All terms are of the form 4k+1, A016813.

If a(n) is never equal to A325319(n), then there are no odd perfect numbers.

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

Cf. A016813, A162296, A228058, A325313, A325319, A325378, A325379.

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(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));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(-A325314(n), ", ")));

a(n) = -A325314(A228058(n)) = A162296(A228058(n)) - A228058(n).
a(n) = A325319(n) - A325379(n) = A325378(n) - A325319(n).
a(n) < A001065(A228058(n)) for all n.

A325319 a(n) = -A325313(A228058(n)).

Original entry on oeis.org

21, 61, 81, 197, 141, 241, 181, 201, 381, 317, 261, 301, 533, 525, 361, 545, 441, 689, 761, 481, 501, 697, 541, 561, 773, 997, 681, 1001, 741, 1305, 781, 1181, 1337, 1153, 861, 1405, 901, 961, 981, 1685, 1509, 1381, 1673, 1841, 1141, 1161, 1201, 2013, 1685, 1281, 2229, 1341, 1837, 1381, 1913, 1401, 2165, 1461, 2501, 2065, 1561, 2141

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

All terms are of the form 4k+1, A016813.

If a(n) is never equal to A325320(n), then there are no odd perfect numbers.

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

Cf. A016813, A048250, A228058, A325313, A325320, A325378, A325379.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - 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));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(-A325313(n), ", ")));

a(n) = -A325313(A228058(n)) = A228058(n) - A048250(A228058(n)).

a(n) = A325320(n) + A325379(n) = A325378(n) - A325320(n).

A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

Original entry on oeis.org

30, 70, 90, 246, 150, 266, 190, 210, 678, 342, 270, 310, 654, 574, 370, 570, 450, 738, 930, 490, 510, 722, 550, 570, 798, 1582, 690, 1026, 750, 2034, 790, 1230, 1626, 1178, 870, 1526, 910, 970, 990, 2046, 1558, 1406, 1722, 1962, 1150, 1170, 1210, 4062, 1710, 1290, 3390, 1350, 1862, 1390, 1938, 1410, 2214, 1470, 3030

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

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

Cf. A048250, A162296, A228058, A325319, A325320, A325379.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
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));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(A162296(n) - A048250(n),", ")));

a(n) = A162296(A228058(n)) - A048250(A228058(n)).

a(n) = A325319(n) + A325320(n).

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).

A325823 Sum of unitary proper divisors of A228058(n): a(n) = A034460(A228058(n)).

Original entry on oeis.org

15, 23, 27, 55, 39, 39, 47, 51, 87, 43, 63, 71, 127, 63, 83, 55, 99, 67, 175, 107, 111, 63, 119, 123, 67, 95, 147, 79, 159, 99, 167, 79, 295, 87, 183, 135, 191, 203, 207, 367, 87, 99, 91, 139, 239, 243, 251, 795, 115, 267, 111, 279, 123, 287, 127, 291, 103, 303, 535, 135, 323, 139, 327, 187, 715, 111, 119, 347, 359, 363, 123, 383

Offset: 1

Antti Karttunen, May 23 2019

nonn

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

Cf. A034448, A034460, A228058, A325379, A325824.

Cf. also A325319, 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
A034460(n) = (A034448(n) - n);
A325823(n) = A034460(A228058(n));

a(n) = A034460(A228058(n)).

a(n) = A325824(n) - A325379(n).

A325310 a(n) = A001511(A325315(n)), except when A325315(n) = 0, then a(n) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

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

Cf. A000396, A001511, A028982 (gives the positions of 1's), A048250, A162296, A228058, A325313, A325314, A325315, A325378, A325379.

Array[If[# == 0, 0, IntegerExponent[2 #, 2]] &[BitXor @@ Abs[#1 - Map[Total, {#3, Complement[#2, #3]}]]] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 105] (* Michael De Vlieger, Apr 21 2019 *)

A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325315(n) = bitxor(abs(A325313(n)),abs(A325314(n)));
A325310(n) = A001511ext(A325315(n));

If A325315(n) = 0, then a(n) = 0, otherwise a(n) = A001511(A325315(n)).

a(A228058(n)) = A001511(abs(A325379(n))), assuming there are no odd perfect numbers, in which case a(A228058(n)) >= 3 for all n.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A228059 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 that are closer to being perfect than previous terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A325320 Sum of proper divisors of A228058(n) that are not squarefree; a(n) = -A325314(A228058(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A325319 a(n) = -A325313(A228058(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs