A091194 -id:A091194 - cp's OEIS frontend

A005101 Abundant numbers (sum of divisors of m exceeds 2m).

12, 18, 20, 24, 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, 258, 260, 264, 270

Offset: 1

N. J. A. Sloane

A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m (A000203).
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!
It appears that for m abundant and > 23, 2*A001055(m) - A101113(m) is NOT 0. - Eric Desbiaux, Jun 01 2009
If m is a term so is every positive multiple of m. "Primitive" terms are in A091191.
If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Thus the n-th abundant number is asymptotic to 4.0322*n < n/A(2) < 4.0421*n. - Daniel Forgues, Oct 11 2015
From Bob Selcoe, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start)
Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:
i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;
ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100);
iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear.
Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].
The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)

L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.
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.
C. K. Caldwell, The Prime Glossary, abundant number.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math., Volume 7, Issue 2 (1998), pp. 137-143.
Jason Earls, On Smarandache repunit n numbers, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), page 243.
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.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
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.
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.
Eric Weisstein's World of Mathematics, Abundant Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Abundant number.
Index entries for "core" sequences.

Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).
Cf. A005231 and A006038 (odd abundant numbers).
Cf. A094268 (n consecutive abundant numbers).
Cf. A173490 (even abundant numbers).
Cf. A001065.
Cf. A000396 (perfect numbers).
Cf. A302991.

a005101 n = a005101_list !! (n-1)
a005101_list = filter (\x -> a001065 x > x) [1..]
-- Reinhard Zumkeller, Nov 01 2015, Jan 21 2013

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d,`,n) fi: od:
isA005101 := proc(n)
    simplify(numtheory[sigma](n) > 2*n) ;
end proc: # R. J. Mathar, Jun 18 2015
A005101 := proc(n)
    option remember ;
    local a ;
    if n =1 then
        12 ;
    else
        a := procname(n-1)+1 ;
        while numtheory[sigma](a) <= 2*a do
            a := a+1 ;
        end do ;
        a ;
    end if ;
end proc: # R. J. Mathar, Oct 11 2017

abQ[n_] := DivisorSigma[1, n] > 2n; A005101 = Select[ Range[270], abQ[ # ] &] (* Robert G. Wilson v, Sep 15 2005 *)
Select[Range[300], DivisorSigma[1, #] > 2 # &] (* Vincenzo Librandi, Oct 12 2015 *)

isA005101(n) = (sigma(n) > 2*n) \\ Michael B. Porter, Nov 07 2009

from sympy import divisors
def ok(n): return sum(divisors(n)) > 2*n
print(list(filter(ok, range(1, 271)))) # Michael S. Branicky, Aug 29 2021

from sympy import divisor_sigma
from itertools import count, islice
def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
A005101_list = list(islice(A005101_gen(), 20)) # Chai Wah Wu, Jan 14 2022

a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - Benoit Cloitre, Sep 04 2002
A005101 = { n | A033880(n) > 0 }. - M. F. Hasler, Apr 19 2012
A001065(a(n)) > a(n). - Reinhard Zumkeller, Nov 01 2015

A169822 Numbers k such that A(k+1) = A(k) + 1, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

1432, 1487, 1849, 2742, 5380, 5434, 6474, 6786, 9752, 10674, 12311, 14115, 14557, 15237, 17266, 17558, 18987, 19138, 19761, 20110, 20259, 20343, 20967, 20997, 22262, 22735, 24342, 25650, 26003, 26471, 27122, 27721, 28914, 28968, 29741, 30203, 30294, 31274, 33322

Offset: 1

N. J. A. Sloane, May 29 2010

nonn

A096399 is the main entry for this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7000 from Muniru A Asiru)

Cf. A005101, A091194, A096399.

A:=Filtered([1..150000],n->Sigma(n)>2*n);;
  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+1); # Muniru A Asiru, Jun 10 2018

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..150000]):
  a:=select(j->A[j+1]=A[j]+1,[$1..nops(A)-1]); # Muniru A Asiru, Jun 10 2018

fQ[n_] := DivisorSigma[1, n] > 2 n; lst = {}; c = 0; k = 1; While[k < 125000, If[fQ@k, c++; If[fQ[k - 1], AppendTo[lst, c - 1]]]; k++ ]; lst (* Robert G. Wilson v, Jun 11 2010 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 1, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

a(n) = A091194(A096399(n)). - Amiram Eldar, Mar 01 2025

a(10) onwards from Robert G. Wilson v, Jun 11 2010

A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

2, 7, 10, 14, 16, 19, 22, 23, 26, 31, 36, 39, 44, 45, 48, 51, 52, 59, 62, 65, 70, 71, 74, 79, 81, 82, 83, 86, 87, 90, 93, 96, 99, 104, 107, 110, 111, 114, 118, 120, 125, 128, 131, 133, 135, 138, 141, 146, 149, 150, 155, 156, 158, 164, 169, 170, 175, 178, 179

Offset: 1

Muniru A Asiru, Jun 22 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)

A231086 is the main entry for this sequence.

Cf. A005101, A091194, A096399, A169822.

A:=Filtered([1..1000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+2);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..1000]):  a:=select(j->A[j+1]=A[j]+2,[$1..nops(A)-1]);

Position[Differences[Select[Range[750], DivisorSigma[1, #] > 2*# &]], 2] // Flatten (* Amiram Eldar, Mar 15 2024 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 2, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { k | A005101(k+1) = A005101(k) + 2 }.

a(n) = A091194(A231086(n)). - Amiram Eldar, Mar 01 2025

A316095 Numbers m such that A(m+1) = A(m) + 3, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

231, 232, 385, 386, 544, 545, 699, 700, 858, 859, 1014, 1015, 1172, 1173, 1326, 1327, 1431, 1488, 1600, 1601, 1645, 1646, 1699, 1700, 1806, 1807, 1850, 1959, 1960, 2015, 2016, 2093, 2094, 2119, 2120, 2221, 2222, 2272, 2273, 2378, 2379, 2433, 2434, 2583, 2584

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

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

A228382 is the main sequence for this entry.
Numbers m such that A(m+1) = A(m) + k, where A() = A005101() are the abundant numbers: A169822 (k=1), A303741 (k=2), this sequence (k=3), A316096 (k=4), A316097 (k=6).
Cf. A005101, A091194.

A:=Filtered([1..20000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+3);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..20000]):  a:=select(j->A[j+1]=A[j]+3,[$1..nops(A)-1]);

Position[Map[{#1, #2 - 3} & @@ # &, Partition[Select[Range[12000], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)

lista(nn) = {my(va = select(x->(sigma(x) > 2*x), [1..nn]), dva = vector(#va-1, k, va[k+1] - va[k])); select(x->(x==3), dva, 1);} \\ Michel Marcus, Jul 03 2018

Sequence is { m | A005101(m+1) = A005101(m) + 3 }.
Sequence is { m | A125115(m) = 3 }.
a(n) = A091194(A228382(n)). - Amiram Eldar, Mar 01 2025

A316096 Numbers m such that A(m+1) = A(m) + 4, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

3, 6, 11, 13, 17, 18, 21, 24, 25, 32, 35, 40, 43, 46, 47, 50, 53, 60, 63, 64, 69, 72, 75, 78, 85, 88, 91, 94, 95, 100, 105, 106, 109, 112, 115, 117, 121, 124, 127, 130, 132, 136, 139, 140, 147, 148, 151, 154, 157, 159, 165, 168, 171, 176, 177, 180, 181, 184

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

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

A316098 is the main sequence for this entry.
Numbers m such that A(m+1) = A(m) + k, where A() = A005101() are the abundant numbers: A169822 (k=1), A303741 (k=2), A316095 (k=3), this sequence (k=4), A316097 (k=6).
Cf. A005101, A091194.

A:=Filtered([1..1000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+4);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..1000]):  a:=select(j->A[j+1]=A[j]+4,[$1..nops(A)-1]);

Position[Map[{#1, #2 - 4} & @@ # &, Partition[Select[Range[10^3], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 4, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { m | A005101(m+1) = A005101(m) + 4 }.
Sequence is { m | A125115(m) = 4 }.
a(n) = A091194(A316098(n)). - Amiram Eldar, Mar 01 2025

A316097 Numbers m such that A(m+1) = A(m) + 6, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

1, 4, 5, 8, 9, 12, 15, 20, 27, 28, 29, 30, 33, 34, 37, 38, 41, 42, 49, 54, 55, 56, 57, 58, 61, 66, 67, 68, 73, 76, 77, 80, 84, 89, 92, 97, 98, 101, 102, 103, 108, 113, 116, 119, 122, 123, 126, 129, 134, 137, 142, 143, 144, 145, 152, 153, 160, 161, 162, 163

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

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

A316099 is the main sequence for this entry.
Numbers m such that A(m+1) = A(m) + k, where A() = A005101() are the abundant numbers: A169822 (k=1), A303741 (k=2), A316095 (k=3), A316096 (k=4), this sequence (k=6).
Cf. A005101, A091194.

A:=Filtered([1..700],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+6);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..700]):  a:=select(j->A[j+1]=A[j]+6,[$1..nops(A)-1]);

Position[Map[{#1, #2 - 6} & @@ # &, Partition[Select[Range[10^3], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 6, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { m | A005101(m+1) = A005101(m) + 6 }.
Sequence is { m | A125115(m) = 6 }.
a(n) = A091194(A316099(n)). - Amiram Eldar, Mar 01 2025

A084840 Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of abundant integers in n-th row.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 2, 2, 2, 3, 3, 4, 4, 4, 2, 4, 4, 4, 7, 5, 4, 6, 5, 7, 6, 8, 7, 6, 9, 7, 8, 9, 8, 9, 9, 8, 10, 10, 10, 10, 10, 12, 11, 9, 12, 11, 11, 14, 10, 14, 14, 13, 14, 13, 13, 14, 16, 15, 13, 16, 15, 17, 17, 15, 18, 14, 17, 16, 19, 18, 19, 15, 20, 19, 18, 20, 20, 18, 20, 21, 22

Offset: 1

Jason Earls, Jun 08 2003

			Triangle begins
1 (0 abundant)
2 3 (0 abundant)
4 5 6 (0 abundant)
7 8 9 10 (0 abundant)
11 12 13 14 15 (1 abundant)
16 17 18 19 20 21 (2 abundant)
22 23 24 25 26 27 28 (1 abundant)

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000217, A005101, A091194, A294937.

f:= proc(n) local i;
      nops(select(t -> numtheory:-sigma(t) > 2*t, [$(n-1)*n/2+1 .. n*(n+1)/2]))
end proc:
map(f, [$1..100]); # Robert Israel, Aug 13 2024

nai[lst_]:=Count[lst,?(DivisorSigma[1,#]-2#>0&)]; Module[{nn=90,tbl},tbl=TakeList[Range[(nn(nn+1))/2],Range[nn]];nai/@tbl] (* _Harvey P. Dale, Jan 27 2025 *)

a(n) = A091194(A000217(n)) - A091194(A000217(n-1)). - Robert Israel, Aug 13 2024

A091195 Number of numbers <= n having only one abundant divisor (A091191).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Reinhard Zumkeller, Dec 27 2003

nonn

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

Partial sums of A294930.

Cf. A091191, A091194, A005101.

f[n_] := Boole[Count[Divisors[n], ?(DivisorSigma[1, #] > 2 # &)] == 1]; Accumulate @ Array[f, 100] (* _Amiram Eldar, Mar 21 2021 *)

A256562 Number of deficient numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 51, 52, 53

Offset: 1

Michel Marcus, Apr 02 2015

nonn

			For k=1 to 5, all numbers are deficients so a(k) = k in this range.
a(6) = 5 since 6 is the first number that is not deficient.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density bounds for the sum of divisors function, Math. Comp. 26 (1972), 773-777.
Eric Weisstein's World of Mathematics, Abundant Number

Partial sums of A294934.
Cf. A000396 (perfect), A005100 (deficient), A005101 (abundant).
Cf. A091194 (number of abundant numbers <= n).
Cf. A256440, A318172.

[#[k:k in [1..n]| DivisorSigma(1,k) lt 2*k]:n in [1..70]]; // Marius A. Burtea, Nov 06 2019

a[n_]:=Length[Select[Range[n],DivisorSigma[1,#]/#<2&]];a/@Range[68] (* Ivan N. Ianakiev, Apr 03 2015 *)

PARI
```
a(n) = sum(k=1, n, sigma(k)/k < 2);
```

a(n) ~ c*n, where c = 0.752380... is the asymptotic density of the deficient numbers (A318172). - Amiram Eldar, Mar 21 2021

A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Original entry on oeis.org

22148167706, 52012, 10828, 24601, 23660, 29114, 58967, 118828, 238600, 478099, 957324, 1916191, 3834167, 7669094, 15335488, 30667762, 61337894, 122679755, 245357929, 490718137, 981456651, 1962956352, 3925957422, 7851819466, 15703524589, 31406984903, 62813576969

Offset: 2

Amiram Eldar, Aug 19 2025

n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.

The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and, respectively.

			a(2) = 22148167706 because there are 22148167706 squarefree numbers k such that A048250(k) > 2*k (i.e., terms of A087248) that are less than the least nonsquarefree number k that has this property, A387154(2) = 401120980260.
a(3) = 52012 because there are 52012 cubefree numbers k such that A073185(k) > 2*k (i.e., terms of A357695) that are less than the least noncubefree number k that has this property, A387154(3) = 360360.

Cf. A004709, A005117, A046099, A048250, A073185, A091194, A302991, A357695, A357700, A387154.

freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];

isfree(n, k) = if(n == 1, 1, my(e = factor(n)[,2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
sigmafree(n, k) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,2],k-1)+1)-1)/(f[i,1]-1));}
a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1;}

Let A_k(n) be the number of k-free abundant numbers that are not exceeding n. Then, a(n) = A_n(A387154(n)) - 1.

a(n) ~ c * 945 * 2^n, where c = A302991.

cp's OEIS Frontend

A005101 Abundant numbers (sum of divisors of m exceeds 2m).

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A169822 Numbers k such that A(k+1) = A(k) + 1, where A() = A005101() are the abundant numbers.

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A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

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A316095 Numbers m such that A(m+1) = A(m) + 3, where A() = A005101() are the abundant numbers.

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A316096 Numbers m such that A(m+1) = A(m) + 4, where A() = A005101() are the abundant numbers.

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A316097 Numbers m such that A(m+1) = A(m) + 6, where A() = A005101() are the abundant numbers.

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A084840 Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of abundant integers in n-th row.