A080224 -id:A080224 - cp's OEIS frontend

A337468 a(n) = A080224(A337386(n)).

7, 8, 10, 7, 13, 10, 13, 12, 12, 12, 8, 9, 18, 9, 17, 13, 8, 16, 7, 17, 9, 7, 19, 8, 9, 7, 17, 18, 16, 9, 23, 7, 10, 18, 16, 16, 6, 24, 9, 21, 15, 9, 13, 19, 17, 22, 15, 17, 26, 15, 9, 15, 9, 17, 22, 9, 29, 9, 23, 19, 9, 15, 16, 9, 14, 28, 17, 11, 17, 25, 17, 23, 14, 22, 9, 14, 25, 16, 16, 31, 17, 15, 15, 11, 20

Offset: 1

Antti Karttunen, Sep 01 2020

nonn

Cf. A000203, A003961, A003973, A080224, A337386.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
A080224(n) = sumdiv(n, d, sigma(d)>2*d)
k=0; for(n=1,2^12,if(isA337386(n),print1(A080224(n),", ")));

a(n) = A080224(A337386(n)).

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

Original entry on oeis.org

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

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

Original entry on oeis.org

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, 246, 258, 272, 282, 304, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822

Offset: 1

Reinhard Zumkeller, Dec 27 2003

nonn

A080224(a(n)) = 1.
This is a supersequence of the primitive abundant number sequence A071395, since many of these numbers will be positive integer multiples of the perfect numbers (A000396). - Timothy L. Tiffin, Jul 15 2016
If the terms of A071395 are removed from this sequence, then the resulting sequence will be A275082. - Timothy L. Tiffin, Jul 16 2016

			12 is a term since 1, 2, 3, 4, and 6 (the proper divisors of 12) are either deficient or perfect numbers, and thus not abundant. - _Timothy L. Tiffin_, Jul 15 2016

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of the abundant numbers, J. London Math. Soc. 9 (1934), pp. 278-282.
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A006038 (odd terms), A005101 (abundant numbers), A091192.
Cf. A027751, A071395 (subsequence), supersequence of A275082.
Cf. A294930 (characteristic function), A294890.

a091191 n = a091191_list !! (n-1)
a091191_list = filter f [1..] 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:
isA091191 := proc(n) local d; if isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if isA005101(d) then return false; end if; end do: return true; else false; end if; end proc:
for n from 1 to 200 do if isA091191(n) then printf("%d\n",n) ; end if;end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 100, n++; If[DivisorSigma[1, n] > 2*n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)
Select[Range@ 840, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* Michael De Vlieger, Jul 16 2016 *)

is(n)=sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Dec 05 2012

Erdős shows that a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 05 2012

A187795 Sum of the abundant divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 20, 0, 0, 0, 36, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 66, 0, 0, 0, 60, 0, 42, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 0, 72, 0, 56, 0, 0, 0, 122, 0, 0, 0, 0, 0, 66, 0, 0, 0, 70, 0, 162, 0, 0, 0, 0, 0, 78, 0, 140, 0, 0, 0, 138, 0, 0, 0, 88, 0, 138, 0, 0, 0, 0, 0, 180

Offset: 1

Timothy L. Tiffin, Jan 06 2013

Sum of divisors d of n with sigma(d) > 2*d.

a(n) = n when n is a primitive abundant number (A091191). - Alonso del Arte, Jan 19 2013

			a(12) = 12 because the divisors of 12 are 1, 2, 3, 4, 6, 12, but of those only 12 is abundant.
a(13) = 0 because the divisors of 13 are 1 and 13, neither of which is abundant.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A000203, A005101, A080224, A270660, A271113, A294889, A294930, A294937.

A187795 := proc(n)
    local a,d;
    a :=0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) > 2* d then
            a := a+d ;
        end if;
    end do:
    return a;
end proc:
seq(A187795(n),n=1..100) ; # R. J. Mathar, Apr 27 2017

Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] > 2 # &], {n, 96}] (* Michael De Vlieger, Jul 16 2016 *)

a(n)=sumdiv(n,d,(sigma(d,-1)>2)*d) \\ Charles R Greathouse IV, Jan 15 2013

from sympy import divisors, divisor_sigma
def A187795(n): return sum(d for d in divisors(n,generator=True) if divisor_sigma(d) > 2*d) # Chai Wah Wu, Sep 22 2021

From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294937(d)*d.
a(n) = A294889(n) + (A294937(n)*n).
If A294889(n) > 0, then a(n) = A294889(n)+n, otherwise a(n) = A294930(n)*n.
a(n) + A187794(n) + A187793(n) = A000203(n).
(End)

A337345 Number of divisors d of n for which A003961(d) > 2*d, where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Number of terms of A246282 that divide n.

Number of divisors d of n for which A048673(d) > d.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Inverse Möbius transform of A252742.
Cf. A003961, A048673, A246282, A337346, A337372 (positions of ones), A341609 (their characteristic function), A341610 (positions of terms > 1), A378658 [= a(A091191(n))], A378662, A378663.
Cf. also A080224, A337541, A341620.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337345(n) = sumdiv(n,d,A003961(d)>(d+d));

a(n) = Sum_{d|n} A252742(d).
a(n) = A337346(n) + A252742(n).
From Antti Karttunen, Dec 10 2024: (Start)
a(n) = 1 <=> A341609(n) = 1.
a(n) = A378662(n) + A080224(n) = A378663(n) + A341620(n).
(End)

A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Antti Karttunen and Peter Munn, Sep 15 2020

nonn

As a simple consequence of the definition of a primitive nondeficient number, a(n) is nonzero if and only if n is nondeficient.

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = a(2) = a(3) = a(4) = a(5) = 0 as all primitive nondeficient numbers are larger, and therefore not divisors; and a(6) = 1, as only 1 primitive nondeficient number divides 6, namely 6 itself.
60 has the following 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Of these, only 6 and 20 are in A006039, thus a(60) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n).

A006039 (or equivalently, its characteristic function, A341619) is used to define this sequence.
See A000203 and A023196 for definitions of deficient and nondeficient.
Sequences with similar definitions: A080224, A294927, A337539, A341620.
Cf. A302110, A341604.
Positions of 0's: A005100.
Positions of numbers >= k: A023196 (k=1), A337688 (k=2), A337689 (k=3).
Positions of first appearances are given in A337691.
Differs from its derived sequence A341618 for the first time at n=120, where a(120)=2, while A341618(120)=1.

A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d= 2*d), return(0))); (1)); \\ After code in A071395
A337690(n) = sumdiv(n, d, A341619(d));

a(n) = Sum_{d|n} A341619(d) = Sum_{d|n} [1==A341620(d)]. - Corrected by Antti Karttunen, Feb 21 2021
a(A005100(n)) = 0.
a(A006039(n)) = 1.
a(A023196(n)) >= 1.
a(A337479(n)) = A337539(n).
a(n) <= A341620(n). - Antti Karttunen, Feb 22 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A006039(n) = 0.3... (see A006039 for a better estimate of this constant). - Amiram Eldar, Jan 01 2024

Data section extended to 120 terms by Antti Karttunen, Feb 21 2021

A080225 Number of perfect divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

Number of divisors d of n with sigma(d) = 2*d (sigma = A000203).

			Divisors of n = 84: {1,2,3,4,6,7,12,14,21,24,28,42}, two of them are perfect: 6 = A000396(1) and 28 = A000396(2), therefore a(84) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Number.
Wikipedia, Perfect number.

Cf. A000203, A000396, A080224, A080226, A147645, A335118.

a080225 n = length [d | d <- takeWhile (<= n) a000396_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 20 2012

a[n_] := DivisorSum[n, 1 &, DivisorSigma[-1, #] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

a(n) = sumdiv(n, d, sigma(d, -1) == 2); \\ Amiram Eldar, Dec 31 2023

A080224(n) + a(n) + A080226(n) = A000005(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A335118 = 0.2045201... . - Amiram Eldar, Dec 31 2023

A080226 Number of deficient divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 4, 5, 2, 4, 2, 5, 4, 4, 2, 5, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 5, 2, 4, 4, 6, 2, 6, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 6, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 7, 2, 6, 2, 4, 6, 6, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 7, 2, 8, 4, 6, 4, 4, 4, 7, 2, 6, 6, 7, 2, 6, 2, 7, 8

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

Number of divisors d of n with sigma(d)<2*d (sigma = A000203).

			All 4 divisors of n=21 are deficient: 1=A005100(1), 3=A005100(3), 7=A005100(6) and 21=A005100(17), therefore a(21)=4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Deficient Number.

Cf. A000203, A005100, A187793, A294926, A294934.

a[n_] := Sum[If[DivisorSigma[1, d] < 2d, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)

A080226(n) = sumdiv(n, d, (sigma(d)<(2*d))); \\ Antti Karttunen, Nov 14 2017

A080224(n) + A080225(n) + a(n) = A000005(n).

a(n) = Sum_{d|n} A294934(d) = A294926(n) + A294934(n). - Antti Karttunen, Nov 14 2017

A341620 Number of nondeficient divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 6, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 8

Offset: 1

Antti Karttunen, Feb 21 2021

nonn

Number of nondeficient numbers (A023196) dividing n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000005, A006039 (positions of ones), A023196, A080224, A080225, A080226, A294927, A294936, A337690, A341619.

Differs from a derived sequence A341624 for the first time at n=120, where a(120)=8, while A341624(120)=1.

a[n_] := DivisorSum[n, 1 &, DivisorSigma[1, #] >= 2*# &]; Array[a, 120] (* Amiram Eldar, Feb 22 2021 *)

A294936(n) = (sigma(n, -1)>=2); \\ From A294936.
A341620(n) = sumdiv(n,d,A294936(d));

A341620(n) = sumdiv(n,d,(sigma(d)>=(2*d)));

a(n) = Sum_{d|n} A294936(d).
a(n) = A294927(n) + A294936(n).
a(n) = A080224(n) + A080225(n) = A000005(n) - A080226(n).
a(n) >= A337690(n) for all n.
a(n) = 1 iff A341619(n) = 1.

A294929 Number of proper divisors of n that are abundant (A005101).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

			The proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. Only one of these, 12, is abundant (in A005101), thus a(24) = 1.
The proper divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60. Six of these are abundant: 12, 20, 24, 30, 40, 60, thus a(120) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005101, A032741, A080224, A294889, A294926, A294927, A294928, A294930, A294937.

Cf. also A294902.

a[n_] := Count[Most[Divisors[n]], ?(DivisorSigma[1, #] > 2*# &)]; Array[a, 100] (* _Amiram Eldar, Mar 14 2024 *)

PARI
```
A294929(n) = sumdiv(n, d, (d(2*d)));
```

a(n) = Sum_{d|n, dA294937(d).
a(n) = A080224(n) - A294937(n).
a(n) + A294928(n) = A032741(n).

cp's OEIS Frontend

A337468 a(n) = A080224(A337386(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

Formula

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A187795 Sum of the abundant divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A337345 Number of divisors d of n for which A003961(d) > 2*d, where A003961 is fully multiplicative with a(p) = nextprime(p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A080225 Number of perfect divisors of n.

Views

Author

Keywords