A006037 -id:A006037 - cp's OEIS frontend

A005100 Deficient numbers: numbers k such that sigma(k) < 2k.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86

Offset: 1

N. J. A. Sloane

A number k is abundant if sigma(k) > 2k (cf. A005101), perfect if sigma(k) = 2k (cf. A000396), or deficient if sigma(k) < 2k (this sequence), where sigma(k) is the sum of the divisors of k (A000203).
Also, numbers k such that A033630(k) = 1. - Reinhard Zumkeller, Mar 02 2007
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Since the perfect numbers have density 0, the deficient numbers have density 0.7520 < 1 - A(2) < 0.7526. Thus the n-th deficient number is asymptotic to 1.3287*n < n/(1 - A(2)) < 1.3298*n. - Daniel Forgues, Oct 10 2015
The data begins with 3 runs of 5 consecutive terms, from 1 to 5, 7 to 11 and 13 to 17. The maximal length of a run of consecutive terms is 5 because 6 is a perfect number and its proper multiples are abundant numbers. - Bernard Schott, May 19 2019
If p and q are primes such that phi(p*q) > p+1, then p*q^n is a term in the sequence for all n >= 1 where phi is the Euler totient function. - Amrit Awasthi, Sep 10 2024

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Britton, Perfect Number Analyser.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), pp. 137-143.
Jose Arnaldo Bebita Dris, A Criterion for Deficient Numbers Using the Abundancy Index and Deficiency Functions, arXiv:1308.6767 [math.NT], 2013-2016; Journal for Algebra and Number Theory Academia, Volume 8, Issue 1 (February 2018), 1-9.
S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 112.
Walter Nissen, Abundancy : Some Resources .
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Eric Weisstein's World of Mathematics, Deficient Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Deficient number.
Index entries for "core" sequences.

Cf. A005101 (abundant), A125499 (even deficient), A247328 (odd deficient), A023196 (complement).
By definition, the weird numbers A006037 are not in this sequence.
Cf. A001065, A318172.

a005100 n = a005100_list !! (n-1)
a005100_list = filter (\x -> a001065 x < x) [1..]
-- Reinhard Zumkeller, Oct 31 2015

with(numtheory); s := proc(n) local i,j,ans; ans := [ ]; j := 0; for i while jA005100 := proc(n)
    numtheory[sigma](n) < 2*n ;
end proc:
A005100 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA005100(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 08 2015

Select[Range[100], DivisorSigma[1, # ] < 2*# &] (* Stefan Steinerberger, Mar 31 2006 *)

isA005100(n) = (sigma(n) < 2*n) \\ Michael B. Porter, Nov 08 2009

for(n=1, 100, if(sigma(n) < 2*n, print1(n", "))) \\  Altug Alkan, Oct 15 2015

from sympy import divisors
def ok(n): return sum(divisors(n)) < 2*n
print(list(filter(ok, range(1, 87)))) # Michael S. Branicky, Aug 29 2021

from sympy import divisor_sigma
from itertools import count, islice
def A005100_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) < 2*n,count(max(startvalue,1))) # generator of terms >= startvalue
A005100_list = list(islice(A005100_gen(),20)) # Chai Wah Wu, Jan 14 2022

A001065(a(n)) < a(n). - Reinhard Zumkeller, Oct 31 2015

More terms from Stefan Steinerberger, Mar 31 2006

A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

Original entry on oeis.org

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955

Offset: 1

N. J. A. Sloane

nonn

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number.
Schiffman notes that 945+630k is in this sequence for all k < 52. Most of the first initial terms are of the form. Among the 1996 terms below 10^6, 1164 terms are of that form, and only 26 terms are not divisible by 5, cf. A064001. - M. F. Hasler, Jul 16 2016
From M. F. Hasler, Jul 28 2016: (Start)
Any multiple of an abundant number is again abundant, see A006038 for primitive terms, i.e., those which are not a multiple of an earlier term.
An odd abundant number must have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. To see this, write the relative abundancy A(N) = sigma(N)/N = sigma[-1](N) as A(Product p_i^e_i) = Product (p_i-1/p_i^e_i)/(p_i-1) < Product p_i/(p_i-1).
See A115414 for terms not divisible by 3, A064001 for terms not divisible by 5, A112640 for terms coprime to 5*7, and A047802 for other generalizations.
As of today, we don't know of a difference between this set S of odd abundant numbers and the set S' of odd semiperfect numbers: Elements of S' \ S would be perfect (A000396), and elements of S \ S' would be weird (A006037), but no odd weird or perfect number is known. (End)
For any term m in this sequence, A064989(m) is also an abundant number (in A005101), and for any term x in A115414, A064989(x) is in this sequence. If there are no odd perfect numbers, then applying A064989 to these terms and sorting into ascending order gives A337386. - Antti Karttunen, Aug 28 2020
There exist infinitely many terms m such that 2*m+1 is also a term. An example of such a term is given by m = 985571808130707987847768908867571007187. - Max Alekseyev, Nov 16 2023

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 13.
R. K. Guy, Unsolved Problems in Number Theory, B2.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

Metin Sariyar, Table of n, a(n) for n = 1..32000 (terms 1..1000 from T. D. Noe)
Jill Britton, Perfect Number Analyzer.
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.
Victor Meally, Letter to N. J. A. Sloane, no date.
Walter Nissen, Abundancy : Some Resources.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.
Jay L. Schiffman and Christopher S. Simons, More Odd Abundant Sequences, Volume 38, Number 1 (September 2005), pp. 7-8.

Cf. A000203, A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725, A064989, A337386.

A005231 := proc(n) option remember ; local a ; if n = 1 then 945 ; else for a from procname(n-1)+2 by 2 do if numtheory[sigma](a) > 2*a then return a; end if; end do: end if; end proc: # R. J. Mathar, Mar 20 2011

fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* Robert G. Wilson v, Mar 20 2011 *)

je=[]; forstep(n=1,15000,2, if(sigma(n)>2*n, je=concat(je,n))); je

is_A005231(n)={bittest(n,0)&&sigma(n)>2*n} \\ M. F. Hasler, Jul 28 2016

list(lim)=my(v=List()); forfactored(n=945,lim\1, if(n[2][1,1]>2 && sigma(n,-1)>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Apr 21 2022

a(n) ~ k*n for some constant k (perhaps around 500). - Charles R Greathouse IV, Apr 21 2022

482.8 < k < 489.8 (based on density bounds by Kobayashi et al., 2009). - Amiram Eldar, Jul 17 2022

More terms from James Sellers

A033630 Number of partitions of n into distinct divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 35, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 2, 1, 7, 1, 1, 1, 26, 1, 1, 1, 2, 1, 24, 1, 1, 1, 1, 1, 22, 1, 1, 1, 3

Offset: 0

Marc LeBrun

nonn

			a(12) = 3 because we have the partitions [12], [6, 4, 2], and [6, 3, 2, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (1000 terms from T. D. Noe)
Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 2.

Cf. A018818, A065205.
Cf. A083206. - Reinhard Zumkeller, Jul 19 2010
Cf. A000009, A005153.
Cf. A211111, A027750.
Cf. A225245.
Cf. A000396, A005100, A005835, A006037.

a033630 0 = 1
a033630 n = p (a027750_row n) n where
   p _  0 = 1
   p [] _ = 0
   p (d:ds) m = if d > m then 0 else p ds (m - d) + p ds m
-- Reinhard Zumkeller, Feb 23 2014, Apr 04 2012, Oct 27 2011

with(numtheory): a:=proc(n) local div, g, gser: div:=divisors(n): g:=product(1+x^div[j],j=1..tau(n)): gser:=series(g,x=0,105): coeff(gser,x^n): end: seq(a(n),n=1..100); # Emeric Deutsch, Mar 30 2006
# second Maple program:
with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n))[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i-1))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014

A033630 = Table[SeriesCoefficient[Series[Times@@((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n ], {n, 512}] (* Wouter Meeussen *)
A033630[n_] := f[n, n, 1]; f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k + 1] + f[n, m - k, k + 1] * Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[A033630, 101, 0] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

a(n) = A065205(n) + 1.
a(A005100(n)) = 1; a(A005835(n)) > 1. - Reinhard Zumkeller, Mar 02 2007
a(n) = f(n, n, 1) with f(n, m, k) = if k <= m then f(n, m, k + 1) + f(n, m - k, k + 1)*0^(n mod k) else 0^m. - Reinhard Zumkeller, Dec 11 2009
a(n) = [x^n] Product_{d|n} (1 + x^d). - Ilya Gutkovskiy, Jul 26 2017
a(n) = 1 if n is deficient (A005100) or weird (A006037). a(n) = 2 if n is perfect (A000396). - Alonso del Arte, Sep 24 2017

More terms from Reinhard Zumkeller, Apr 21 2003

A005835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264

Offset: 1

N. J. A. Sloane

In other words, some subset of the numbers { 1 <= d < n : d divides n } adds up to n. - N. J. A. Sloane, Apr 06 2008
Also, numbers n such that A033630(n) > 1. - Reinhard Zumkeller, Mar 02 2007
Deficient numbers cannot be pseudoperfect. This sequence includes the perfect numbers (A000396). By definition, it does not include the weird, i.e., abundant but not pseudoperfect, numbers (A006037).
From Daniel Forgues, Feb 07 2011: (Start)
The first odd pseudoperfect number is a(233) = 945.
An empirical observation (from the graph) is that it seems that the n-th pseudoperfect number would be asymptotic to 4n, or equivalently that the asymptotic density of pseudoperfect numbers would be 1/4. Any proof of this? (End)
A065205(a(n)) > 0; A210455(a(n)) = 1. - Reinhard Zumkeller, Jan 21 2013
Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers (A005101) and weird numbers (A006037), since the remaining integers are perfect numbers (A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers (A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790. - Jaycob Coleman, Oct 26 2013
The odd terms of this sequence are given by the odd abundant numbers A005231, up to hypothetical (so far unknown) odd weird numbers (A006037). - M. F. Hasler, Nov 23 2017
The term "pseudoperfect numbers" was coined by Sierpiński (1965). The alternative term "semiperfect numbers" was coined by Zachariou and Zachariou (1972). - Amiram Eldar, Dec 04 2020

			6 = 1+2+3, 12 = 1+2+3+6, 18 = 3+6+9, etc.
70 is not a member since the proper divisors of 70 are {1, 2, 5, 7, 10, 14, 35} and no subset adds to 70.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 144.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Anonymous, Semiperfect Numbers: Definition [Broken link]
Stan Benkoski, Problem E2308, Amer. Math. Monthly, Vol. 79, No. 7 (1972), p. 774.
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., Vol. 28, No. 126 (1974), pp. 617-623. Corrigendum, Math. Comp., Vol. 29, No. 130 (1975), pp. 673-674.
David Eppstein, Is it known whether a group of Egyptian fractions with odd, distinct denominators can add up to 1?, 1996.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
Wacław Sierpiński, Sur les nombres pseudoparfaits, Matematički Vesnik, Vol. 2 (17), No. 33 (1965), pp. 212-213.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; arXiv:math preprint, arXiv:math/1812.06566 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Semiperfect Number.
Wikipedia, Semiperfect number.
Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.

Subsequence of A023196; complement of A136447.
See A136446 for another version.
Cf. A006036, A005100, A033630, A000396, A005231.
Cf. A109761 (subsequence).

a005835 n = a005835_list !! (n-1)
a005835_list = filter ((== 1) . a210455) [1..]
-- Reinhard Zumkeller, Jan 21 2013

with(combinat):
isA005835 := proc(n)
    local b, S;
    b:=false;
    S:=subsets(numtheory[divisors](n) minus {n});
    while not S[finished] do
        if convert(S[nextvalue](), `+`)=n then
            b:=true;
            break
        end if ;
    end do;
    b
end proc:
for n from 1 do
    if isA005835(n) then
        print(n);
    end if;
end do: # Walter Kehowski, Aug 12 2005

A005835 = Flatten[ Position[ A033630, q_/; q>1 ] ] (* Wouter Meeussen *)
pseudoPerfectQ[n_] := Module[{divs = Most[Divisors[n]]}, MemberQ[Total/@Subsets[ divs, Length[ divs]], n]]; A005835 = Select[Range[300],pseudoPerfectQ] (* Harvey P. Dale, Sep 19 2011 *)
A005835 = {}; n = 0; While[Length[A005835] < 100, n++; d = Most[Divisors[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[A005835, n]]]; A005835 (* T. D. Noe, Dec 29 2011 *)

is_A005835(n, d=divisors(n)[^-1], s=vecsum(d), m=#d)={ m||return; while(d[m]>n, s-=d[m]; m--||return); d[m]==n || if(nA005835(n-d[m], d, s-d[m], m-1) || is_A005835(n, d, s-d[m], m-1), n==s)} \\ Returns nonzero iff n is the sum of a subset of d, which defaults to the set of proper divisors of n. Improved using more recent PARI syntax by M. F. Hasler, Jul 15 2016, Jul 27 2016. NOTE: This function is also used (with 2nd optional arg) in A136446, A122036 and possibly in A006037. - M. F. Hasler, Jul 28 2016
for(n=1,1000,is_A005835(n)&&print1(n",")) \\ M. F. Hasler, Apr 06 2008

Better description and more terms from Jud McCranie, Oct 15 1997

A023196 Nondeficient numbers: numbers k such that sigma(k) >= 2k; union of A000396 and A005101.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252

Offset: 1

David W. Wilson

Also called the non-deficient numbers.
If k is a term, so is every positive multiple of k. The "primitive" terms form A006039.
The sequence of numbers k that give local minima for A004125, i.e., such that A004125(k-1) > A004125(k) and A004125(k) < A004125(k+1) coincides with this sequence for the first 1014 terms. Then there appears 4095 which is a term of A023196 but is not a local minimum. - Max Alekseyev, Jan 26 2005
Also, union of pseudoperfect and weird numbers. Cf. A005835, A006037. - Franklin T. Adams-Watters, Mar 29 2006
k is in the sequence if and only if A004125(k-1) > A004125(k). - Jaycob Coleman, Mar 31 2014
Like the abundant numbers, this sequence has density between 0.2474 and 0.2480, see A005101. - Charles R Greathouse IV, Nov 30 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B (2016) Vol. 3, 1-26.

Cf. A000203, A004125, A006039, A000396, A005101.

Filtered([1..260],n->Sigma(n)>=2*n); # Muniru A Asiru, Dec 04 2018

[n: n in [1..300] | not (2*n gt DivisorSigma(1,n))]; // Vincenzo Librandi, Dec 05 2018

A023196:=n->`if`(numtheory[sigma](n)>=2*n, n, NULL): seq(A023196(n), n=1..380); # Wesley Ivan Hurt, Apr 18 2017

Select[Range[300], DivisorSigma[1, #] >= 2# &] (* Harvey P. Dale, Sep 26 2014 *)

is(n)=sigma(n,-1)>=2 \\ Charles R Greathouse IV, Mar 09 2014

A002975 Primitive weird numbers: weird numbers with no proper weird divisors.

Original entry on oeis.org

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

Offset: 1

N. J. A. Sloane

Sidney Kravitz notes that a(21) = 539744; it was misprinted as 539774 in the Benkoski & Erdős article. - Charles R Greathouse IV, Apr 04 2012
It appears that a weird number is primitive iff, divided by its largest prime factor, it is not weird. Is there a simple proof for this? - M. F. Hasler, Aug 20 2014 [The comment below does not answer this question.]
Yes, any primitive weird number, pwn, multiplied by any prime > sigma_1(pwn) is also weird. - Robert G. Wilson v, Jun 09 2015
A proper subsequence of A006037 and A091191. - Robert G. Wilson v, May 25 2015
Number of terms < 10^n: 0, 1, 2, 7, 13, 24, 48, 85, 152, 276, 499, 881, ..., . - Robert G. Wilson v, Jun 21 2017
The primitive weird number (pwn) 176405960704 is the least term which has as its abundance a pwn. Two other terms are 81152249741312, 14327148694372352. - Robert G. Wilson v, Sep 22 2017
Primitive weird numbers == 2 (mod 4): {70, 4030, 5830, 4199030, 1550860550, 66072609790, ...}. All the terms in A258374 appear so far. - Robert G. Wilson v, Nov 21 2015
See A258882 (and A258333) for terms of the form a(n)=2^k*p*q and A258401 for all other terms, with subsets A258883 (a(n)=2^k*p*q*r), A258884 (a(n)=2^k*p*q*r*s), A258885 (six distinct prime factors). A258374 and A258375 list the smallest terms with n prime factors (with / without counting multiplicity). - M. F. Hasler, Jul 12 2016
Sequence A273815 lists terms with nonsquarefree odd part, by definition excluded in A258883 and A258884. - M. F. Hasler, Feb 18 2018
Let n be a weird number and d be a divisor of n. If n/d is not weird, then either it is deficient or it is pseudoperfect. But if n/d is pseudoperfect, then multiplying the subset of the divisors of n/d that sums to n/d by d gives a solution for n, contradicting the assumption that n is weird. Therefore, n/d must be deficient. Of all the prime factors of n contributing to sigma(n)/n, the largest prime will contribute the least, and so if n/gpf(n) is deficient, then n/d is deficient for all divisors d of n, and n is a primitive weird number. - Charlie Neder, Oct 08 2018
The second part of the above reasoning is incorrect: gpf(n) may contribute more to sigma(n)/n than a smaller prime factor. For example, for n = 24, we have n/3 deficient, but n/2 abundant; for n = 350, n/7 is deficient but n/5 is abundant. - M. F. Hasler, Jan 25 2020

			10430 = A006037(8) is weird but not primitive weird because it has the proper weird divisor 70 = A006037(1).

R. K. Guy, Unsolved Problems in Number Theory, B2.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1161 (terms a(58)-a(160) from Donovan Johnson). [Term a(1159) inserted and b-file reformatted by _Georg Fischer_, Jan 16 2019]
Stan Benkoski, Problem E2308, Amer. Math. Monthly, 79 (1972) 774.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi and Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi and Maurizio Parton, Primitive abundant and weird numbers with many prime factors, Journal of Number Theory vol. 201 (2019) pp. 436-459. DOI: 10.1016/j.jnt.2019.02.027. (Preprint: arXiv:1802.07178.)
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
G. Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.
Robert G. Wilson v, Table of n, a(n) along with its abundance and their factorizations for n = 1..1161

Cf. A006037.

Cf. A258882 and A258333; A258883, A258884, A258885 and A258401; A258374 and A258375.

(* first do *) << Combinatorica` (* then *) fQ[n_] := Block[{d = Most@ Divisors@ n, l = 2^(DivisorSigma[0, n] - 1), i = 1}, i = 1; While[i < l && Plus @@ NthSubset[i, d] != n, i++ ]; i == l]; lst = {}; Do[m = n; If[ Mod[n, 6] != 0 && DivisorSigma[1, n] > 2 n && Union[ Mod[ n, Join[lst, {n + 1}]]][[1]] != 0 && fQ@n, AppendTo[lst, n]; Print@n], {n, 2, 42000000, 2}] (* Robert G. Wilson v, Aug 04 2009 *)
(* Input: Range of even numbers --- Output: Primitive weird numbers *)
Block[{$RecursionLimit = Infinity},
  subOfSum[ss_, kk_, rr_] :=
   Module[{s = ss, k = kk, r = rr},
    If[s + w[[k]] >= mm && s + w[[k]] <= m, t = False;
     Goto[done] (* Found *),
     If[s + w[[k]] + w[[k + 1]] <= m,
      subOfSum[s + w[[k]], k + 1, r - w[[k]]]];
     If[s + r - w[[k]] >= m && s + w[[k + 1]] <= m,
      subOfSum[s, k + 1, r - w[[k]] ]]]; t]; (* end subOfSum *)
  greedyQ[ab_] := Module[{abn = ab, v, sum, s, j, jj, k}, tt = False;
    jj = Length[w]; (* start search *)
    Do[s = r; sum = 0; Do[v = w[[j]]; sum = sum + v;
      If[sum > abn, sum = sum - v; Goto[nxt]];
      If[sum == abn, tt = True; Goto[doneG]]; s = s - v;
      Label[nxt], {j, jj, 1, -1}];
     jj = jj - 1, {k, 1, jj - 1}]; Label[doneG];
    (* True means found, False not found *) tt]; (* end greedyQ *)
  cnt = 0;
  Do[ If[ Mod[n, 3] == 0, Goto[agn]]; r = DivisorSigma[1, n];
   m = r - 2*n;
   If[m > 0, fi = FactorInteger[n]; largestP = fi[[Length[fi]]][[1]];
    nn = n/largestP; If[m > 2*nn || Length[fi] < 3, Goto[agn]];
    If[DivisorSigma[1, nn] > 2*nn, Goto[agn]]; t = True; r = r - n;
    ww = Divisors[n]; lenW = Length[ww];
    Do[ If[ ww[[i]] <= m, w = Drop[ww, i - lenW]; Break[],
      r = r - ww[[i]]], {i, lenW - 1, 1, -1}];
    If[r >= m,
     If[ greedyQ[m], t = False, (* Powers of 2 dropped *)
      exp2 = fi[[1]][[2]]; sig2 = 2^(exp2 + 1) - 1; mm = m - sig2;
      lenW = Length[w]; ww = {};
      If[exp2 > 1,
       Do[ Do[ If[ w[[i]] == 2^ii, ww = AppendTo[ww, w[[i]]]],
      {i, 1, lenW}], {ii, 0, exp2}];
        w = Complement[w, ww]
       (* end T if *), w = Drop[w, 2]];
      (* end Pwr2 *) t = subOfSum[0, 1, r]]]; Label[done];
    If[t, Print[++cnt, "   ", n, "  ", t]]];
   Label[agn], {n, 2, 10000000, 2}]]
(* from Brent Baughn via http://mathematica.stackexchange.com/questions/73301/calculating-weird-numbers, Robert G. Wilson v, Nov 21 2015 *)

is_A002975(n)=is_A006037(n)&&!fordiv(n,d,!bittest(d,0)&&dA006037(d)&&return) \\ M. F. Hasler, Jan 07 2014

More terms from Jud McCranie, Oct 21 2001
One more term from Robert G. Wilson v, Aug 04 2009
a(1)-a(123) double-checked by M. F. Hasler, Jan 07 2014
Edited by M. F. Hasler, Jul 12 2016

A006038 Odd primitive abundant numbers.

Original entry on oeis.org

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215

Offset: 1

N. J. A. Sloane

nonn

Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. Sequence A188342 lists the smallest such numbers. - T. D. Noe, Mar 28 2011
Sequence A188439 sorts the numbers in this sequence by the number of distinct prime factors. Eight numbers have exactly three prime factors; 576 have exactly four prime factors. - T. D. Noe, Apr 04 2011
Any multiple of an abundant number (A005101) is again an abundant number. Primitive abundant numbers (A091191) are those not of this form, i.e., without an abundant proper divisor. We don't know any odd perfect number (A000396), so the (odd) terms here have only deficient proper divisors (A071395), and their prime factors p are less than sigma(n/p)/deficiency(n/p). See A005231 (odd abundant numbers) for an explanation why all terms have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. All known terms are semiperfect (A005835, and thus in A006036): no odd weird number (A006037) is known, but if it exists, the smallest one is in this sequence. - M. F. Hasler, Jul 28 2016
So far, a(173) = 351351 is the only known term of A122036, i.e., which can't be written as sum of its proper divisors > 1. - M. F. Hasler, Jan 26 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jacob Liddy, An algorithm to determine all odd primitive abundant numbers with d prime divisors, Honors Research Projects (2018), 728.
Eric Weisstein's World of Mathematics, Primitive Abundant Number

Cf. A005101, A005231. Subsequence of A091191.

Cf. A000203, A027751, A379949 (subsequence of square terms).

a006038 n = a006038_list !! (n-1)
a006038_list = filter f [1, 3 ..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA005100 := proc(n) is(numtheory[sigma](n) < 2*n ); end proc:
isA006038 := proc(n) local d; if type(n,'odd') and isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if not isA005100(d) then return false; end if; end do: return true;else false; end if; end proc:
n := 1 ; for a from 1 by 2 do if isA006038(a) then printf("%d %d\n",n,a) ; n := n+1 ; end if; end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 50, n = n + 2; If[DivisorSigma[1, n] > 2 n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)

is(n)=n%2 && sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Jun 10 2013

is_A006038(n)=bittest(n,0) && sigma(n)>2*n && !for(i=1,#f=factor(n)[,1],sigma(n\f[i],-1)>2&&return) \\ More than 5 times faster. - M. F. Hasler, Jul 28 2016

A064771 Let S(n) = set of divisors of n, excluding n; sequence gives n such that there is a unique subset of S(n) that sums to n.

Original entry on oeis.org

6, 20, 28, 78, 88, 102, 104, 114, 138, 174, 186, 222, 246, 258, 272, 282, 304, 318, 354, 366, 368, 402, 426, 438, 464, 474, 490, 496, 498, 534, 572, 582, 606, 618, 642, 650, 654, 678, 748, 762, 786, 822, 834, 860, 894, 906, 940, 942, 978, 1002, 1014, 1038

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001

Perfect numbers (A000396) are a proper subset of this sequence. Weird numbers (A006037) are numbers whose proper divisors sum to more than the number, but no subset sums to the number.
Odd elements are rare: the first few are 8925, 32445, 351351, 442365; there are no more below 100 million. See A065235 for more details.
A065205(a(n)) = 1. - Reinhard Zumkeller, Jan 21 2013

			Proper divisors of 20 are 1, 2, 4, 5 and 10. {1,4,5,10} is the only subset that sums to 20, so 20 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1..200 from T. D. Noe, terms 201..5000 from Amiram Eldar)

A005835 gives n such that some subset of S(n) sums to n. Cf. A065205.
Cf. A006037, A065205, A378448 (characteristic function).
Subsequences: A000396, A065235 (odd terms), A378519, A378530.
Cf. A027751.

a064771 n = a064771_list !! (n-1)
a064771_list = map (+ 1) $ elemIndices 1 a065205_list
-- Reinhard Zumkeller, Jan 21 2013

filter:= proc(n)
  local P,x,d;
  P:= mul(x^d+1, d = numtheory:-divisors(n) minus {n});
  coeff(P,x,n) = 1
end proc:
select(filter, [$1..2000]); # Robert Israel, Sep 25 2024

okQ[n_]:= Module[{d=Most[Divisors[n]]}, SeriesCoefficient[Series[ Product[ 1+x^i, {i, d}], {x, 0, n}], n] == 1];Select[ Range[ 1100],okQ] (* Harvey P. Dale, Dec 13 2010 *)

from sympy import divisors
def isok(n):
    dp = {0: 1}
    for d in divisors(n)[:-1]:
        u = {}
        for k in dp.keys():
            if (s := (d + k)) <= n:
                u[s] = dp.get(s, 0) + dp[k]
                if s == n and u[s] > 1:
                    return False
        for k,v in u.items():
            dp[k] = v
    return dp.get(n, 0) == 1
print([n for n in range(1, 1039) if isok(n)]) # Darío Clavijo, Sep 17 2024

More terms from Don Reble, Jud McCranie and Naohiro Nomoto, Oct 22 2001

cp's OEIS Frontend

A306951 Number of weird numbers (A006037) below 10^n.

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A362916 Least weird number (A006037) with exactly n divisors that are weird numbers, or -1 if no such number exists.

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A005100 Deficient numbers: numbers k such that sigma(k) < 2k.

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A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

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A033630 Number of partitions of n into distinct divisors of n.

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A005835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.

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A023196 Nondeficient numbers: numbers k such that sigma(k) >= 2k; union of A000396 and A005101.