A006036 -id:A006036 - cp's OEIS frontend

A335290 Primitive pseudoperfect numbers (A006036) that are not primitive abundant (A071395).

6, 28, 350, 490, 496, 770, 910, 1190, 1330, 1610, 2030, 2170, 2590, 2870, 3010, 3290, 3710, 4130, 4270, 4690, 4970, 5110, 5530, 5810, 6230, 6790, 7070, 7210, 7490, 7630, 7910, 8128, 8890, 9170, 9196, 9590, 9730, 15884, 19228, 24244, 25916, 30932, 34276, 35948

Offset: 1

Amiram Eldar, May 30 2020

nonn

Includes all the perfect numbers (A000396). The nonperfect terms have an abundant proper divisor which is not pseudoperfect, i.e., a proper divisor which is a weird number (A006037).
The first term with one weird divisor is a(3) = 350, having the weird divisor 70.
The first term with 2 weird divisors is a(202) = 658312, having the 2 weird divisors 9272 and 10792.
The first term with 3 weird divisors is a(353) = 1574930, having the 3 weird divisors 70, 10430 and 10570.

			350 is a term since it is pseudoperfect: 1 + 5 + 14 + 35 + 50 + 70 + 175 = 350. All of its proper divisors, {1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175} are not pseudoperfect, and it is not primitive abundant, since its divisor 70 is abundant.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A000396, A006036, A006037, A071395, A306987.

pspQ[n_] := Module[{d = Most @ Divisors[n], x}, Plus @@d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; primPspQ[n_] := pspQ[n] && AllTrue[Most @ Divisors[n], !pspQ[#] &]; primAbQ[n_] := DivisorSigma[1, n] > 2*n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2*# &]; Select[Range[1000], primPspQ[#] && !primAbQ[#] &]

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

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

A308710 Primitive practical numbers of the form 2^i * prime(k).

Original entry on oeis.org

6, 20, 28, 88, 104, 272, 304, 368, 464, 496, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 4288, 4544, 4672, 5056, 5312, 5696, 6208, 6464, 6592, 6848, 6976, 7232, 8128, 16768, 17536, 17792, 19072, 19328, 20096, 20864, 21376, 22144, 22912, 23168, 24448, 24704, 25216

Offset: 1

Miko Labalan, Jun 19 2019

nonn

Intersection of A267124 and A100368.
a(n) is a number of the form 2^i * prime(k) for i > 0 and A007053(i) < k <= A007053(i+1).
Terms are pseudoperfect numbers, A005835 and are also primitive pseudoperfect numbers, A006036.

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

Cf. A000040, A000079, A005153, A005835, A006036, A007053, A100368, A267124.

a[n_] := (p = Prime[n+1]) * 2^Floor[Log2[p]]; Array[a, 50] (* Amiram Eldar, Sep 22 2019 *)

isok(n) = is_A100368(n) && is_A267124(n); \\ Michel Marcus, Jun 19 2019

a(n) = 2^floor(log_2(prime(n+1))) * prime(n+1).

A306987 Primitive abundant numbers (A071395) that are pseudoperfect (A005835).

Original entry on oeis.org

20, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4070, 4095, 4216, 4288, 4408, 4510, 4544, 4672

Offset: 1

Amiram Eldar, Mar 18 2019

nonn

By definition these numbers are also primitive pseudoperfect (A006036).

Benkoski and Erdős proved that this sequence is infinite, since it includes all the numbers of the form 2^k * p with p a prime such that 2^k < p < 2^(k+1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617-623. Corrigendum: Math. Comp. 29 (1975), 673-674.

Cf. A005835, A006036, A071395.

paQ[n_]:=DivisorSigma[1, n] > 2n && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ n] == 1; psQ[n_]:=Module[{d= Most[Divisors[n] ]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; Select[Range[5000], paQ[#]&&psQ[#]&] (* after Michael De Vlieger at A071395 and T. D. Noe at A005835 *)

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]

A109762 Rare primitive pseudoperfect numbers: primitive pseudoperfect numbers k such that k == 2 or 10 (mod 12).

Original entry on oeis.org

350, 490, 550, 650, 770, 910, 1190, 1330, 1430, 1610, 1870, 2002, 2030, 2090, 2170, 2210, 2470, 2530, 2590, 2870, 2990, 3010, 3190, 3230, 3290, 3410, 3710, 3770, 4070, 4130, 4270, 4510, 4690, 4730, 4970

Offset: 1

Walter Kehowski, Aug 13 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3000

Cf. A006036, A109761.

PSP := proc(l::list) local x, R, S, P, L; S:=sort(l); R:=[]; P:=S; for x in S do if not(x in R) then L:=selectremove(proc(z) not(z=x) and z mod x = 0 end, P); R:=[op(R),op(L[1])]; P:=L[2]; fi; od; return P; end: PSP(RSP); # RSP is sequence of rare pseudoperfect numbers

A192285 Primitive pseudo anti-perfect numbers.

Original entry on oeis.org

5, 7, 8, 17, 22, 23, 31, 33, 38, 39, 41, 52, 53, 59, 67, 71, 73, 74, 81, 83, 94, 101, 103, 108, 109, 116, 122, 127, 129, 137, 143, 149, 151, 157, 158, 167, 171, 172, 178, 179, 193, 199, 214, 237, 241, 247, 257, 262, 263, 269, 283, 293, 311, 313, 319, 331, 333

Offset: 1

Paolo P. Lava, Jul 20 2011

nonn

A primitive pseudo anti-perfect number is a pseudo anti-perfect number that is not a multiple of any other pseudo anti-perfect number.
Like A006036 but using anti-divisors.
Subset of A192270.

Cf. A006036, A066272, A192270

with(combinat);
P:=proc(i)
local a,j,k,n,ok,S,v;
v:=array(1..10000); j:=0;
for n from 1 to i do
  a:={};
  for k from 2 to n-1 do
    if abs((n mod k)- k/2) < 1 then a:=a union {k}; fi;
  od;
  S:=subsets(a);
  while not S[finished] do
    if convert(S[nextvalue](), `+`)=n then
       if j=0 then j:=1; v[1]:=n; print(n); break;
       else
          ok:=1;
          for k from 1 to j do
              if trunc(n/v[k])=n/v[k] then ok:=0; break; fi;
          od;
          j:=j+1; v[j]:=n; if ok=1 then print(n); fi;
       fi;
    fi;
  od;
od;
end:

A367481 Primitive practical numbers of the form 2 * 3^i * prime(k).

Original entry on oeis.org

30, 42, 66, 78, 306, 342, 414, 522, 558, 666, 2214, 2322, 2538, 2862, 3186, 3294, 3618, 3834, 3942, 4266, 4482, 4806, 5238, 5454, 5562, 5778, 5886, 6102, 20574, 21222, 22194, 22518, 24138, 24462, 25434, 26406, 27054, 28026, 28998, 29322, 30942, 31266, 31914

Offset: 1

Miko Labalan, Nov 19 2023

nonn

This sequence and A308710 are both non-overlapping subsets of A267124.
a(n) is a number of the form 2 * 3^i * prime(k) for i > 0 and b(i) < k <= b(i+1) where b(n) = Sum_{m=2..n+1} A233919(m).
Terms are pseudoperfect numbers, A005835, but are not primitive pseudoperfect numbers, A006036.
Since no term is a square or twice a square, there are no terms k such that sigma(k) is odd. Therefore, according to Proposition 10 by Rao/Peng (see their JNT paper at A083207) all terms are Zumkeller numbers. - Ivan N. Ianakiev, Nov 28 2023

Cf. A000040, A000244, A005153, A005835, A006036, A233919, A267124, A308710.

a[n_]:=2*3^(Floor[Log[2*Prime[n+2]]/Log[3]]-1)*Prime[n+2]; Array[a,43] (* Stefano Spezia, Nov 19 2023 *)

a(n) = 2 * 3^(floor(log_3(2*prime(n+2)))-1) * prime(n+2).

cp's OEIS Frontend

A335290 Primitive pseudoperfect numbers (A006036) that are not primitive abundant (A071395).

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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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A006038 Odd primitive abundant numbers.

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A308710 Primitive practical numbers of the form 2^i * prime(k).

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A306987 Primitive abundant numbers (A071395) that are pseudoperfect (A005835).

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

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Mathematica

A109762 Rare primitive pseudoperfect numbers: primitive pseudoperfect numbers k such that k == 2 or 10 (mod 12).

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Maple

A192285 Primitive pseudo anti-perfect numbers.

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