A091193 -id:A091193 - cp's OEIS frontend

A080224 Number of abundant divisors of n.

0, 0, 0, 0, 0, 0, 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, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

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

a(n)>0 iff n is abundant: a(A005101(n))>0, a(A000396(n))=0 and a(A005100(n))=0; a(A091191(n))=1; a(A091192(n))>1; a(A091193(n))=n and a(m)<>n for m < A091193(n). - Reinhard Zumkeller, Dec 27 2003

			Divisors of n=24: {1,2,3,4,6,8,12,24}, two of them are abundant: 12=A005101(1) and 24=A005101(4), therefore a(24)=2.

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

Cf. A000005, A000203, A005101, A080225, A080226, A187795, A294890, A294929, A294930, A294937.

Cf. also A294904.

A080224 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) > 2*d then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A080224(n),n=1..80) ; # R. J. Mathar, Feb 22 2021

Table[Count[Divisors[n],?(DivisorSigma[1,#]>2#&)],{n,110}] (* _Harvey P. Dale, Jun 14 2013 *)

a(n) = sumdiv(n, d, sigma(d)>2*d)  \\ Michel Marcus, Mar 09 2013

a(n) + A080225(n) + A080226(n) = A000005(n).
From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294937(d).
a(n) = A294929(n) + A294937(n).
a(n) = 1 iff A294930(n) = 1.
(End)

A335540 Numbers with a record number of abundant divisors.

Original entry on oeis.org

1, 12, 24, 36, 60, 72, 120, 180, 240, 360, 720, 1080, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 151200, 166320, 221760, 277200, 302400, 332640, 443520, 554400

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

The corresponding numbers of abundant divisors are 0, 1, 2, 3, 4, 5, 7, 8, 10, 13, 18, 19, 23, ...
All the terms > 1 are abundant numbers (A005101) and all the terms > 12 are not primitive abundant numbers (A091191).
Apparently, all the terms are least numbers of their prime signature (A025487). This was verified for the first 100 terms.

			12 is in the sequence since it is the least number with one abundant divisor (12). The next number with more than one abundant divisor is 24 which has 2 abundant divisors (12 and 24).

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000203, A005101, A025487, A080224, A091191, A091193, A335541, A335542.

s[n_] := Count[Divisors[n], _?(DivisorSigma[1, #] > 2*# &)]; sm = -1; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

Numbers m such that A080224(m) > A080224(k) for all k < m.

A337691 a(n) is the least positive integer divisible by exactly n primitive nondeficient numbers (A006039).

Original entry on oeis.org

1, 6, 60, 140, 420, 3780, 17160, 28600, 40040, 138600, 120120, 180180, 300300, 360360, 600600, 1351350, 900900, 4144140, 1801800, 3063060, 5405400, 6126120, 8558550, 7657650, 19399380, 20720700, 17117100, 15315300, 29099070, 30630600, 45945900, 70450380, 91891800, 87297210

Offset: 0

David A. Corneth, Antti Karttunen and Peter Munn, Sep 20 2020

nonn

a(10) starts a run of at least 31 terms divisible by 30030 = 13#, product of primes <= 13.

About 20% of known terms are not divisible by 4 (indices 0, 1, 15, 22, 23, 28, 33, 38, 40, ...). This contrasts with many sequences that require terms to have some higher measure of abundancy (cf. A002093, A004394, A004490), where almost all terms are divisible by 4. The possibility of nontrivial odd terms seems worth considering.

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = 6, as the smallest number divisible by exactly 1 primitive nondeficient number.
Table of n, a(n) and the relevant divisors starts:
  n    a(n)   divisors in A006039
  0       1   (none);
  1       6   6;
  2      60   6, 20;
  3     140   20, 28, 70;
  4     420   6, 20, 28, 70;
  5    3780   6, 20, 28, 70, 945;
  6   17160   6, 20, 88, 104, 572, 1430;
  7   28600   20, 88, 104, 550, 572, 650, 1430;
  8   40040   20, 28, 70, 88, 104, 572, 1430, 2002; ...
Note that a(6), a(7), a(8) are 3*5720, 5*5720, 7*5720.

David A. Corneth, Table of n, a(n) for n = 0..43
David A. Corneth, Some more upper bounds on a(n) for n at most 110.

A006039, A337690 are used to define this sequence.
See A000203 and A023196 for definitions of deficient and nondeficient.
Sequences with similar definitions: A091193, A335540, A338405.
Cf. A002093, A004394, A004490.

\\ Code for A337690 given under that entry.
A337691list(search_up_to_n) = { my(m=Map(),lista=List([]),t); for(n=1,search_up_to_n,if(!(n%(2^24)),print1("(",n,")")); t=A337690(n); if(!mapisdefined(m,t), mapput(m,t,n))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v337691 = A337691list(2^27);
A337691(n) = v337691[1+n];

a(n) = min({k integer : k >= 1 and A337690(k) = n}).

cp's OEIS Frontend

A080224 Number of abundant divisors of n.

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A335540 Numbers with a record number of abundant divisors.

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A337691 a(n) is the least positive integer divisible by exactly n primitive nondeficient numbers (A006039).

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