A088831 -id:A088831 - cp's OEIS frontend

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

250, 376, 1276, 12616, 20536, 396916, 801376, 1297312, 8452096, 33721216, 40575616, 59376256, 89397016, 99523456, 101556016, 150441856, 173706136, 269096704, 283417216, 500101936, 1082640256, 1846506832, 15531546112, 34675557856, 136310177392, 136783784608

Offset: 1

Max Alekseyev, Aug 27 2025

Contains numbers 2^(k-1)*(2^k + 31) for k in A247952.

Max Alekseyev, Table of n, a(n) for n = 1..54

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A247952.

A259174 Numbers whose abundance is a power of 2.

Original entry on oeis.org

12, 20, 56, 70, 88, 104, 108, 220, 368, 464, 550, 572, 650, 748, 836, 860, 952, 992, 1232, 1504, 1672, 1888, 1952, 2140, 2392, 2744, 3708, 4030, 5336, 5830, 6328, 6536, 6808, 7192, 7304, 7544, 7912, 8968, 9656, 9820, 10184, 10792, 11096, 13496, 14008

Offset: 1

Robert G. Wilson v, Jun 20 2015

Subsequence of A005101 whose abundance is a term of A000079 except 1.
Below 35*10^8, only 236925 is odd and its abundance is 2^9.
Least terms with abundance 2^e for e = 1, 2, ... are listed in A292558.

Robert G. Wilson v, Table of n, a(n) for n = 1..482

Contains as subsequences A088831, A088832, A088833, A141547, A175989, A275996, A292626.

Cf. A005101, A292558.

fQ[n_] := IntegerQ@ Log2[DivisorSigma[1, n] - 2 n]; Select[ Range@ 15000, fQ]

isok(n) = isprimepower(sigma(n)-2*n, &p) && (p==2); \\ Michel Marcus, Mar 25 2017

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]

A281626 a(n) = (sum of trivial divisors of n) - (sum of nontrivial divisors of n).

Original entry on oeis.org

1, 3, 4, 3, 6, 2, 8, 3, 7, 4, 12, -2, 14, 6, 8, 3, 18, -1, 20, 0, 12, 10, 24, -10, 21, 12, 16, 2, 30, -10, 32, 3, 20, 16, 24, -17, 38, 18, 24, -8, 42, -10, 44, 6, 14, 22, 48, -26, 43, 9, 32, 8, 54, -10, 40, -6, 36, 28, 60, -46, 62, 30, 24, 3, 48, -10, 68, 12

Offset: 1

Jaroslav Krizek, Feb 11 2017

sign

Trivial divisors of n are numbers 1 and n.
a(n) = 0 for numbers in A088831 (numbers n whose abundance is 2).
a(n) <= 0 for numbers in A005101 (abundant numbers).
a(n) > 0 for numbers in A263837 (non-abundant numbers).

			a(20) = (20+1) - (2+4+5+10) = 0.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000079, A000396, A005101, A006881, A048050, A088831, A263837.

[1] cat [2*(n+1) - SumOfDivisors(n): n in [2..100]];

Table[If[n == 1, 1, 2 (n + 1) - DivisorSigma[1, n]], {n, 68}] (* Michael De Vlieger, Feb 11 2017 *)

a(1) = 1; for n>1, a(n) = (n+1) - (sigma(n) - n - 1) = 2*(n+1) - sigma(n) = n + 1 - A048050(n).
a(A000396(n)) = 2 for n >= 1.
a(A000079(n)) = 3 for n >= 1.
a(A006881(n)) = phi(n).
a(p) = p + 1 for p prime.

A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

Original entry on oeis.org

15, 20, 35, 95, 104, 119, 143, 207, 209, 287, 319, 323, 377, 464, 527, 559, 650, 779, 899, 923, 989, 1007, 1023, 1189, 1199, 1343, 1349, 1519, 1763, 1919, 1952, 2015, 2159, 2507, 2759, 2911, 2915, 2975, 3239, 3599, 3827, 4031, 4199, 4607, 5183, 5207, 5249

Offset: 1

Zdenek Cervenka, Oct 23 2017

nonn

Numbers k such that sigma(k)/(sigma(k)-k-1) is a positive integer.

			15 is in the sequence since sigma(15)/(sigma(15)-15-1) = 24/8 = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002808 (composite numbers).
Cf. A000203, A048050.
Cf. A088831 (k=2), A063906 (k=3).

Quiet@ Select[Range[2, 5300], And[IntegerQ[#], # > 1] &[#2/(#2 - #1 - 1)] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Oct 24 2017 *)

lista(nn) = forcomposite(n=1, nn, if (denominator(sigma(n)/(sigma(n)-n-1)) == 1, print1(n, ", "))); \\ Michel Marcus, Oct 24 2017

list(lim)=my(v=List(),s,t); forfactored(n=9,lim\1, s=sigma(n); t=s-n[1]-1; if(t && s%t==0, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 11 2017

This sequence gives all numbers a(n) in increasing order which satisfy A000203(a(n))/A048050(a(n)) = A000203(a(n))/(A000203(a(n)) - (a(n)+1)) = k(n), with a positive integer k(n) for n >= 1. - Wolfdieter Lang, Nov 10 2017

Edited by Wolfdieter Lang, Nov 10 2017

Name corrected by Michel Marcus, Nov 12 2017

A322162 Numbers k such that bsigma(k) = 2k + 2, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

80, 104, 832, 1952, 7424, 62464, 522752, 8382464, 33357824, 134193152, 267649024, 17167286272, 549754241024

Offset: 1

Amiram Eldar, Nov 29 2018

The bi-unitary version of A088831.

If m is a term of A050414, i.e., 2^m - 3 is prime, then 2^(2*m-2) * (2^m-3) is in this sequence, and also 2^(m-1) * (2^m-3) if m is even.

			80 is in this sequence since its sum of bi-unitary divisors is 162 = 2 * 80 + 2.

Cf. A050414, A088831, A188999.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; Select[Range[2,10000], Times@@(fun @@@ FactorInteger[#]) == 2#+2 &]

bsigma(n,f=factor(n))=prod(i=1,#f~, my(p=f[i,1], e=f[i, 2]); if (e%2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)));
forfactored(n=1,10^8, if(bsigma(n[1],n[2])==2*n[1]+2, print1(n[1]", "))) \\ Charles R Greathouse IV, Nov 29 2018

a(13) from Giovanni Resta, Dec 01 2018

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A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

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A259174 Numbers whose abundance is a power of 2.

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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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A281626 a(n) = (sum of trivial divisors of n) - (sum of nontrivial divisors of n).

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A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

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A322162 Numbers k such that bsigma(k) = 2k + 2, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

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