A156903 -id:A156903 - cp's OEIS frontend

A204824 Duplicate of A156903.

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 968, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4624, 4900, 5000

Offset: 1

dead

A171641 Non-deficient numbers with even sigma which are not Zumkeller.

Original entry on oeis.org

738, 748, 774, 846, 954, 1062, 1098, 1206, 1278, 1314, 1422, 1494, 1602, 1746, 1818, 1854, 1926, 1962, 2034, 2286, 2358, 2466, 2502, 2682, 2718, 2826, 2934, 3006, 3114, 3222, 3258, 3438, 3474, 3492, 3546, 3582, 3636, 3708, 3798, 3852, 3924, 4014, 4068, 4086

Offset: 1

Peter Luschny, Dec 14 2009

nonn

Numbers which are non-deficient (sigma(n) >= 2n) [A023196] such that sigma(n) [A000203] is even but which are not Zumkeller numbers [A083207], i.e., the positive factors of n cannot be partitioned into two disjoint parts so that the sums of the two parts are equal.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Chai Wah Wu)
Peter Luschny, Zumkeller Numbers.

Cf. A000203, A023196, A083207.
Setwise difference A083211 \ A156903.
Positions of even negative terms in A378600.

with(NumberTheory):
isA171641 := proc(n) local s, p, i, P;
    s := SumOfDivisors(n);
    if s::odd or s < n*2 then false else
    P := mul(1 + x^i, i in Divisors(n));
    0 = coeff(P, x, s/2) fi end:
select(isA171641, [seq(1..4100)]);  # Peter Luschny, Oct 19 2024

Reap[For[n = 2, n <= 4000, n = n+2, sigma = DivisorSigma[1, n]; If[sigma >= 2n && EvenQ[sigma] && Coefficient[ Times @@ (1 + x^Divisors[n]) // Expand, x, sigma/2] == 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 26 2013 *)

from sympy import divisors
import numpy as np
A171641 = []
for n in range(2,10**6):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and 2*n <= s:
        d.remove(n)
        s2, ld = int(s/2-n), len(d)
        z = np.zeros((ld+1,s2+1),dtype=int)
        for i in range(1,ld+1):
            y = min(d[i-1],s2+1)
            z[i,range(y)] = z[i-1,range(y)]
            z[i,range(y,s2+1)] = np.maximum(z[i-1,range(y,s2+1)],z[i-1,range(0,s2+1-y)]+y)
        if z[ld,s2] != s2:
            A171641.append(n)
# Chai Wah Wu, Aug 19 2014

A156942 Odd abundant numbers whose abundance is odd.

Original entry on oeis.org

11025, 99225, 245025, 275625, 342225, 540225, 893025, 1334025, 1863225, 2205225, 2480625, 3080025, 3186225, 3980025, 4601025, 4862025, 5832225, 6125625, 6890625, 7868025, 8037225, 8555625, 9272025, 9828225, 10595025, 10989225

Offset: 1

Robert G. Wilson v, Feb 18 2009

nonn

Number of terms <10^n: 0, 0, 0, 0, 2, 7, 24, 83, 250, 792, 2484, 7988, 25383, 80082, ..., . Not all are a multiple of 25, i.e.; 81162081 = 9009^2 = (9*7*11*13)^2. See A156943.
Any term must be an odd square. Square roots are in A174830.
Indeed, the sum of divisors of any number isn't odd unless it's a square or twice a square (A028982), and to get the abundance, twice the number is subtracted, so the parity remains the same. - M. F. Hasler, Jan 26 2020
Question: Is this a subsequence of A379503? (Is A379504(a(n)) > 0 for all n? See A379951). The first 15000 terms are all included there. - Amiram Eldar and Antti Karttunen, Jan 06 2025
Question 2: Is A379505(a(n)) > 1 for all n, especially if there are no quasiperfect numbers (numbers k such that sigma(k) = 2k+1)? - Antti Karttunen, Jan 06 2025
From Amiram Eldar, Jan 16 2025: (Start)
The least term that is not divisible by 5 is a(75) = 81162081.
The least term that is not divisible by 3 is a(296889) = 1382511906801025.
The least term that is coprime to 15 is 15285071557677427358507559514565648611799881. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A005101, A005231, A028982, A033880, A156903, A174830, A379503, A379504, A379505, A379951 [= A379504(a(n))].

Subsequences: A156943, A325311 (thus also A379490), A347890, A379949 (terms that are primitive abundant).

fQ[n_] := Block[{ds = DivisorSigma[1, n] - 2 n}, ds > 0 && OddQ@ ds]; Select[ Range[1, 12006223, 2], fQ @# &]

is(n)=my(s=sigma(n)); n%2 && s>2*n && (s-2*n)%2 \\ Charles R Greathouse IV, Feb 21 2017

a(n) = A174830(n)^2. - M. F. Hasler, Jan 26 2020

Edited by Robert G. Wilson v at the suggestion of T. D. Noe, Mar 30 2010

A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

Original entry on oeis.org

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 738, 748, 774, 784, 800, 846, 882, 900, 954, 968, 1062, 1098, 1152, 1206, 1278, 1296, 1314, 1352, 1422, 1458, 1494, 1568, 1600, 1602, 1746, 1764, 1800, 1818, 1854, 1926, 1936, 1962, 2034, 2178, 2286, 2304, 2358, 2450, 2466, 2500, 2502, 2592, 2682, 2704, 2718, 2826, 2916, 2934, 3006, 3042

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

A083206(a(n)) = 0; subsequence of A083210.

All [abundant] numbers with an odd sum of divisors (either a square or twice a square, A028982) must be terms because for these numbers the two subsets will be of opposite parity. - Robert G. Wilson v, Apr 01 2010, clarified by Antti Karttunen, Dec 05 2024

			Divisors of n=18: {1,2,3,6,9,18}; 18 is pseudo-perfect (A005835): 18=9+6+3, but there exist no two complementary subsets of divisors having the same sum, therefore 18 is a term.

David A. Corneth, Table of n, a(n) for n = 1..12915 (first 6061 terms from Antti Karttunen)
Eric Weisstein's World of Mathematics, Abundant Number.
Reinhard Zumkeller, Illustration of initial terms

Intersection of A005101 and A083210.
Disjoint union of A156903 and A171641. - Amiram Eldar, Jun 20 2020
Positions of negative terms in A378600.
Cf. A000203, A028982, A083206, A156942 (odd terms), A378661 (characteristic function).

fQ[n_] := Block[{d = Divisors[n], t, ds, x}, ds = Total[d]; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[3042], And[DivisorSigma[1, #] > 2 #, ! fQ[#]] &] (* Michael De Vlieger, Dec 04 2024, after T. D. Noe at A083207 *)

A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); };
is_A083211(n) = ((sigma(n)>2*n) && (0==A083206(n))); \\ Antti Karttunen, Dec 04 2024

{k such that sigma(k) > 2*k and A083206(k) = 0}. - Antti Karttunen, Dec 04 2024

a(21)-a(46) from Robert G. Wilson v, Apr 01 2010

Many missing terms inserted, first ones at a(29) = 1206 and a(30) = 1278 - Antti Karttunen, Dec 04 2024

A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 0, 12, 4, 6, 1, 16, -1, 18, 0, 10, 8, 22, 0, 19, 10, 14, 0, 28, 0, 30, 1, 18, 14, 22, -1, 36, 16, 22, 0, 40, 0, 42, 4, 12, 20, 46, 0, 41, 7, 30, 6, 52, 0, 38, 0, 34, 26, 58, 0, 60, 28, 22, 1, 46, 0, 66, 10, 42, 0, 70, -1, 72, 34, 26, 12, 58, 0, 78, 0, 41, 38, 82, 0, 62, 40, 54, 0, 88, 0, 70

Offset: 1

Antti Karttunen, Dec 04 2024

sign

If n is abundant, then negate the value of A103977(n), otherwise use as it is.

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

Cf. A033879, A103977.

Cf. A005100 (positions of terms > 0), A083207 (positions of 0's), A083211 (positions of negative terms), A156903 (positions of odd negative terms), A171641 (of even negative terms).

A033879(n) = (n+n-sigma(n));
nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); };
A103977(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); my(plist=nonzerocoefpositions(p), m = #plist, d); if(!(m%2), plist[1+(m/2)]-plist[m/2], d = plist[(m+1)/2]-plist[(m-1)/2]; if(1==d,0,d)); };
A378600(n) = { my(d=A033879(n)); if(d>=0, d, -A103977(n)); };

If A033879(n) >= 0, a(n) = A033879(n), otherwise a(n) = -A103977(n).

A204825 Abundant numbers with even sum of divisors.

Original entry on oeis.org

12, 20, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Numbers m from A005101 such that sigma(m) = A000203(m) is even.

Complement of A156903 with respect to A005101 (abundant numbers).

			Abundant number 12 is in sequence because sigma(12) = 28 (even number).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A156903 (abundant numbers with odd sum of divisors), A204826 (deficient numbers with odd sum of divisors), A204827 (deficient numbers with even sum of divisors).

aQ[n_] := EvenQ[(s = DivisorSigma[1, n])] && s > 2n; Select[Range[280], aQ] (* Amiram Eldar, Sep 02 2019 *)

A204826 Deficient numbers with odd sum of divisors.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 32, 49, 50, 64, 81, 98, 121, 128, 169, 225, 242, 256, 289, 338, 361, 441, 484, 512, 529, 578, 625, 676, 722, 729, 841, 961, 1024, 1058, 1089, 1156, 1225, 1250, 1369, 1444, 1521, 1681, 1682, 1849, 1922, 2025, 2048, 2116, 2209, 2312, 2401

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Numbers m from A005100 such that sigma(m) = A000203(m) is odd.

Complement of A204827 with respect to A005100 (deficient numbers).

			Deficient number 16 is in sequence because sigma(16) = 13 (odd number).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deficient Number

Cf. A000203, A005100.

Cf. A156903 (abundant numbers with odd sum of divisors), A204825 (abundant numbers with even sum of divisors), A204827 (deficient numbers with even sum of divisors).

aQ[n_] := OddQ[(s = DivisorSigma[1, n])] && s < 2n; Select[Range[2401], aQ] (* Amiram Eldar, Sep 02 2019 *)

A204827 Deficient numbers with even sum of divisors.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 103

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Numbers m from A005100 such that sigma(m) = A000203(m) is even.

Complement of A204826 with respect to A005100 (deficient numbers).

			Deficient number 15 is in sequence because sigma(15) = 24 (even number).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deficient Number

Cf. A000203, A005100, A028983.

Cf. A156903 (abundant numbers with odd sum of divisors), A204825 (abundant numbers with even sum of divisors), A204826 (deficient numbers with odd sum of divisors).

dnesQ[n_]:=Module[{s=DivisorSigma[1,n]},s<2n&&EvenQ[s]]; Select[ Range[ 120],dnesQ] (* Harvey P. Dale, Nov 23 2014 *)

A140863 Odd numbers k such that sigma(m) = 2m+k has a solution in m, where sigma is the sum-of-divisors function A000203.

Original entry on oeis.org

3, 7, 17, 19, 31, 39, 41, 51, 59, 65, 71, 89, 115, 119, 127, 161, 185, 199, 215, 243, 251, 259, 265, 269, 299, 309, 353, 363, 399, 401, 455, 459, 467, 499, 519, 593, 635, 713, 737, 815, 831, 845, 899, 921, 923, 965, 967, 983, 1011, 1021, 1025, 1049, 1053, 1055

Offset: 1

Lekraj Beedassy, Jul 20 2008

nonn

From M. F. Hasler and Farideh Firoozbakht, Nov 26 2009: (Start)
The sequence of Mersenne primes, A000668 is a subsequence of this sequence.
Because if k=2^p-1 is prime then n=2^(p-1)*(2^p-1)^2 is a solution of the equation sigma(x)=2x+k. The proof is easy. (End)
The definition is equivalent to asking for a number m with abundance A033880(m) = k. - M. F. Hasler, Mar 10 2025

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.

Robert G. Wilson v, Table of n, a(n) for n = 1..579
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A000668. - M. F. Hasler and Farideh Firoozbakht, Nov 26 2009
Cf. A156903. - Robert G. Wilson v, Dec 09 2018
Cf. A000203 (sigma), A033880 (abundance: sigma(n)-2n).
Cf. A380866 (smallest solutions m to the given equation).

A033880(A156903), image of A156903 under A033880, or range of A033880 restricted to A156903, where A033880 is the abundance sigma(x)-2x, and A156903 are numbers with odd positive abundance. - M. F. Hasler, Mar 10 2025

a(13)-a(54) from Donovan Johnson, Dec 09 2008

A156943 Odd abundant numbers not divisible by 25 whose abundancy is odd.

Original entry on oeis.org

81162081, 138791961, 173369889, 441882441, 730458729, 1091179089, 1249127649, 1524043521, 1560329001, 1744649361, 2286465489, 2606204601, 3255501249, 3976941969, 4770526761, 5567099769, 6574128561, 6800806089, 7584145569

Offset: 1

Robert G. Wilson v, Feb 18 2009

nonn

Number of terms < 10^n: 0, 0, 0, 0, 0, 0, 0, 1, 5, 25, 63, ..., .

Robert G. Wilson v, Table of n, a(n) for n=1..1005

Subsequence of A156903 and A156942.

q[n_] := Module[{ab = DivisorSigma[1, n] - 2*n}, ab > 0 && OddQ[ab]]; Select[Range[1, 7775536039, 2], ! Divisible[#, 25] && q[#] &] (* corrected by Amiram Eldar, Oct 10 2023 *)

Name corrected by Amiram Eldar, Oct 10 2023

cp's OEIS Frontend

A204824 Duplicate of A156903.

Views

Author

Keywords

A171641 Non-deficient numbers with even sigma which are not Zumkeller.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A156942 Odd abundant numbers whose abundance is odd.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A204825 Abundant numbers with even sum of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A204826 Deficient numbers with odd sum of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A204827 Deficient numbers with even sum of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs