A136446 -id:A136446 - cp's OEIS frontend

A122036 Odd abundant numbers (A005231) which are not in A136446, i.e., not sum of some of their proper divisors > 1.

351351

Offset: 1

N. J. A. Sloane, Apr 11 2008, following correspondence from R. K. Guy, M. F. Hasler and others

It is conjectured that there are no odd weird numbers (A006037), i.e., that all odd abundant numbers (A005231) are pseudoperfect (A005835); this sequence lists those which are not equal to the sum of a subset of proper divisors > 1.
No second term in the range <= 53850001. - R. J. Mathar, Mar 21 2011
No other terms congruent to 21 (mod 30) below 10^9. - M. F. Hasler, Jul 16 2016
a(2) > 10^16. - Wenjie Fang, Jul 17 2017

			a(1) = 351351 = 3^3 * 7 * 11 * 13^2 is the sum of all its 47 proper divisors (including 1) except 7 and 11, but it is not possible to get the same sum without using the trivial divisor 1: The sum of all proper divisors *larger than 1* yields 351351 + 7 + 11 - 1 = 351351 + 17, and it is not possible to get 17 as sum of a subset of {3, 7, 9, 11, 13, 21, ...}. Thus, 351351 is not in A136446, and therefore in this sequence. - _M. F. Hasler_, Jul 16 2016, edited Mar 15 2021

Index entries for one-term sequences

Cf. A005231, A005835, A006037, A136446.

is_A122036(n)={n>351350 && !is_A005835(n,n=divisors(n)[2..-2]) && n && vecsum(n)>=n[1]*n[#n] && n[1]>2} \\ (Checking for abundant & odd after is_A005835() rather than before, to make it faster when operating on candidates known to satisfy these conditions.) Updated for current PARI syntax by M. F. Hasler, Jul 16 2016, further edits Jan 31 2020
forstep(n=1,10^7,2, is_A122036(n) && print1(n","))

Comments and PARI code from M. F. Hasler, Apr 12 2008

Edited by M. F. Hasler, Jul 16 2016, Mar 15 2021

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

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

A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

Original entry on oeis.org

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670

Offset: 1

N. J. A. Sloane

OProject@Home in subproject Weird Engine calculates and stores the weird numbers.
There are no odd weird numbers < 10^17. - Robert A. Hearn (rah(AT)ai.mit.edu), May 25 2005
From Alois P. Heinz, Oct 30 2009: (Start)
The first weird number that has more than one decomposition of its divisors set into two subsets with equal sum (and thus is not a member of A083209) is 10430:
  1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215
  2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215. (End)
There are no odd weird numbers < 1.8*10^19. - Wenjie Fang, Sep 04 2013
S. Benkowski and P. Erdős (1974) proved that the asymptotic density W of weird numbers is positive. It can be shown that W < 0.0101 (see A005835). - Jaycob Coleman, Oct 26 2013
No odd weird number exists below 10^21. This search was done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014
No odd weird number with abundance less than 10^14 exists below 10^28. See Odd Weird Search link. - Wenjie Fang, Feb 25 2015
A weird number k multiplied by a prime p > sigma(k) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor. Sequence A065235 lists odd numbers that can be written in only one way as sum of their divisors, and A122036 lists those which are not in A136446, i.e., not sum of proper divisors > 1. - M. F. Hasler, Jul 30 2016

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 70, p. 24, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 129.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 4901 terms from Lukasz Swierczewski)
Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.
S. Benkoski, Are All Weird Numbers Even?, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.
David Eppstein, Eqyptian Fractions.
Wenjie Fang, Searching on the boundary of abundance for odd weird numbers, arXiv:2207.12906 [math.NT], 2022.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.
Odd Weird Search, Report on the recently completed batch, Feb 23 2015.
OProject, Weird numbers list.
J. Sandor and B. Crstici, Handbook of number theory II, chapter 1.8. [Broken link]
Eric Weisstein's World of Mathematics, Weird Number.
Wikipedia, Weird number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1992.
Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993.

Cf. A002975, A005101, A005835, A005100, A138850, A087167.

Cf. A210455.

a006037 n = a006037_list !! (n-1)
a006037_list = filter ((== 0) . a210455) a005101_list
-- Reinhard Zumkeller, Jan 21 2013

isA006037 := proc(n)
    isA005101(n) and not isA005835(n) ;
end proc:
for n from 1 do
    if isA006037(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Jun 18 2015

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) fQ[n_] := Block[{d, l, t, i}, If[ DivisorSigma[1, n] > 2n && Mod[n, 6] != 0, d = Take[Divisors[n], {1, -2}]; l = 2^Length[d]; t = Table[ NthSubset[j, d], {j, l - 1}]; i = 1; While[i < l && Plus @@ t[[i]] != n, i++ ]]; If[i == l, True, False]]; Select[ Range[ 20000], fQ[ # ] &] (* Robert G. Wilson v, May 20 2005 *)

is_A006037(n,d=divisors(n),s=vecsum(d)-n,m=#d-1)={ m||return; while(d[m]>n, s-=d[m]; m--); d[m]n, is_A006037(n-d[m], d, s-d[m], m-1) && is_A006037(n, d, s-d[m], m-1), sM. F. Hasler, Mar 30 2008; improved and updated to current PARI syntax by M. F. Hasler, Jul 15 2016

is_A006037(n, d=divisors(n)[^-1], s=vecsum(d))={s>n && !is_A005835(n,d,s)} \\ Equivalent but slightly faster than the self-contained version above.-- For efficiency, ensure that the argument is even or add "!bittest(n,0) && ..." to check this first. - M. F. Hasler, Jul 17 2016

t=0; A006037=vector(100,i, until( is_A006037(t+=2),); t) \\ M. F. Hasler, Mar 30 2008

More terms from Jud McCranie, Oct 21 2001

A211111 Number of partitions of n into distinct divisors > 1 of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 19, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 14, 1

Offset: 0

Reinhard Zumkeller, Apr 01 2012

nonn

a(A136446(n)) > 1.

			n=12: the divisors > 1 of 12 are {2,3,4,6,12}, there are exactly two subsets which sum up to 12, namely {12} and {2,4,6}, therefore a(12) = 2;
a(13) = #{13} = 1, because 13 is prime, having no other divisor > 1;
n=14: the divisors > 1 of 14 are {2,7,14}, {14} is the only subset summing up to 14, therefore a(14) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..1000 from Reinhard Zumkeller)

Cf. A211110, A033630, A027750.

Cf. A065205, A136446.

a211111 n = p (tail $ a027750_row n) n where
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m | m < k     = 0
               | otherwise = p ks (m - k) + p ks m

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
      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, Nov 18 2021

a[n_] := Count[IntegerPartitions[n, All, Divisors[n] // Rest], P_ /; Reverse[P] == Union[P]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 18 2021 *)

a(0)=1 prepended by Alois P. Heinz, Nov 18 2021

A065235 Odd numbers which can be written in precisely one way as sum of a subset of their proper divisors.

Original entry on oeis.org

8925, 32445, 351351, 442365, 159427275, 159587925, 159677175, 159784275, 159837825, 159855675, 159944925, 159962775, 160016325, 160105575, 160266225, 160284075, 160391175, 160444725, 160480425, 160533975, 160551825, 160766025, 161015925, 161033775, 161069475

Offset: 1

Jud McCranie, Oct 23 2001

nonn

From Antti Karttunen, Nov 28 2024: (Start)
Characteristic function of this sequence is c(n) = A000035(n)*A378448(n).
The only non-multiples of 25 among the first 10000 terms are a(2)..(4): 32445 = 3^2 * 5 * 7 * 103, 351351 = 3^3 * 7 * 11 * 13^2 and 442365 = 3 * 5 * 7 * 11 * 383, while the other 9997 terms are all of the form 25 * some squarefree number. No terms of A228058 occur among the initial 10000 terms. Compare also to A348743.
(End)

			See A064771 for an example when the number is even.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences where any odd perfect numbers must occur

Odd terms in A064771 (a unique subset of proper divisors sums to the number).

Cf. A000035, A033630, A065205, A136446, A228058, A348743, A378448.

{k such that k is odd and A065205(k) = 1}. - Antti Karttunen, Nov 28 2024

Definition clarified by M. F. Hasler, Apr 08 2008

More terms from Giovanni Resta, Oct 04 2019

A342398 Numbers k such that there is a subset of the nontrivial unitary divisors of k, {d|k : 1 < d < k, gcd(d, k/d) = 1}, that adds up to k.

Original entry on oeis.org

30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834, 840, 858

Offset: 1

Amiram Eldar, Mar 10 2021

nonn

			30 is a term since its proper unitary divisors, 1 < d < 30, are {2, 3, 5, 6, 10, 15}, and 5 + 10 + 15 = 30.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary version of A136446.

Subsequence of A034683 and A293188.

q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] > 0]; Select[Range[1000], q]

A339343 Abundant pseudoperfect numbers k such that no subset of the nontrivial divisors {d|k : 1 < d < k} sums to k.

Original entry on oeis.org

20, 88, 104, 272, 304, 350, 368, 464, 572, 650, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 3770, 4288, 4544, 4672, 5056, 5312, 5696, 5704, 5810, 6208, 6464, 6592, 6790, 6808, 6848, 6976, 7144, 7232, 7630, 7910, 8024, 8056, 9590, 9730, 10744, 11096, 11288, 13192

Offset: 1

Amiram Eldar, Nov 30 2020

nonn

Numbers that are the sum of a proper subset of their aliquot divisors but are not the sum of any subset of their nontrivial divisors.
The perfect numbers (A000396) which are a subset of the pseudoperfect numbers (A005835) are excluded from this sequence since otherwise they would all be trivial terms: if k is a perfect number then the sum of the divisors {d|k : 1 < d < k} is k-1, so any subset of them has a sum smaller than k.
The pseudoperfect numbers are thus a disjoint union of the perfect numbers, this sequence, and A136446.
The abundant numbers (A005101) are a disjoint union of the weird numbers (A006037), this sequence, and A136446.
All the terms are primitive pseudoperfect (A006036), since if k*m is a pseudoperfect number with k > 1, and m also pseudoperfect, then it is a sum of a subset of its divisors, all of which are multiples of k and therefore larger than 1.
This sequence is infinite. If p is an odd prime that is not a Mersenne prime (A000668), and k is the least number such that 2^k * p is an abundant number (A005101; i.e., the least k such that 2^(k+1) - 1 > p), then 2^k * p is a term (these are the nonperfect terms of A308710). If 2^k * p was not a term, then since it has only 2 odd divisors (1 and p), it would be equal to a sum of its even divisors (if 1 is not in the sum then p also cannot be in it). This would make 2^(k-1) * p also a pseudoperfect number, but by definition of k, 2^(k-1) * p is a deficient number (A005100).
If k is an even abundant number with abundance (A033880) 2, i.e., sigma(k) = A000203(k) = 2*k + 2, then k is a term.
a(157) = A122036(1) = 351351 is the least (and currently the only known) odd term.

			20 is a term since it is a pseudoperfect number, 20 = 1 + 4 + 5 + 10, and the set of nontrivial divisors of 20, {d|20 : 1 < d < 20} = {2, 4, 5, 10}, has no subset that sums to 20.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000396, A000668, A005100, A005101, A005835, A006037, A033880, A070824, A088831, A122036, A136446, A308710.
Subsequence of A006036.
A122036 is a subsequence.

psQ[n_] := DivisorSigma[1, n] > 2*n && Module[{d = Most@Divisors[n], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0 ]; Select[Range[2000], psQ]

cp's OEIS Frontend

A122036 Odd abundant numbers (A005231) which are not in A136446, i.e., not sum of some of their proper divisors > 1.

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

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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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A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

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A211111 Number of partitions of n into distinct divisors > 1 of n.

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A065235 Odd numbers which can be written in precisely one way as sum of a subset of their proper divisors.

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