A080226 -id:A080226 - 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)

A187793 Sum of the deficient divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 10, 14, 24, 24, 31, 18, 15, 20, 22, 32, 36, 24, 18, 31, 42, 40, 28, 30, 36, 32, 63, 48, 54, 48, 19, 38, 60, 56, 30, 42, 48, 44, 84, 78, 72, 48, 34, 57, 93, 72, 98, 54, 42, 72, 36, 80, 90, 60, 40, 62, 96, 104, 127, 84, 72, 68, 126, 96, 74, 72, 27

Offset: 1

Timothy L. Tiffin, Jan 06 2013

Sum of divisors d of n with sigma(d) < 2*d.
a(n) = sigma(n) when n is itself also deficient.
Also, a(n) agrees with the terms in A117553 except when n is a multiple (k > 1) of either a perfect number or a primitive abundant number.
Notice that a(1) = 1. The remaining fixed points are given by A125310. - Timothy L. Tiffin, Jun 23 2016
a(A028982(n)) is an odd integer.  Also, if n is an odd abundant number that is not a perfect square and n has an odd number of abundant divisors (e.g., 945 has one abundant divisor and 4725 has three abundant divisors), then a(n) will also be odd: a(945) = 975 and a(4725) = 2675. - Timothy L. Tiffin, Jul 18 2016

			a(12) = 10 because the divisors of 12 are 1, 2, 3, 4, 6, 12; of these, 1, 2, 3, 4 are deficient, and they add up to 10.

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

Cf. A000203, A005100, A028982, A080226, A117553, A125310, A125499, A187794, A187795, A247328, A274338, A274339, A274340, A274380, A274549, A274829, A294886, A294934.

A187793 := 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:
    a ;
end proc:# R. J. Mathar, May 08 2019

Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &], {n, 72}] (* Michael De Vlieger, Jul 18 2016 *)

a(n)=sumdiv(n,d,if(sigma(d,-1)<2,d,0)) \\ Charles R Greathouse IV, Jan 07 2013

From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294934(d)*d.
a(n) = A294886(n) + (A294934(n)*n).
a(n) + A187794(n) + A187795(n) = A000203(n).
(End)

a(54) corrected by Charles R Greathouse IV, Jan 07 2013

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

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.

A294926 Number of proper divisors of n that are deficient (A005100).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

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

Cf. A032741, A080226, A294886, A294927, A294928, A294929, A294934.

a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] < 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

A294926(n) = sumdiv(n, d, (dAntti Karttunen, Nov 14 2017

a(n) = Sum_{d|n, dA294934(d).
a(n) = A080226(n) - A294934(n).
a(n) + A294927(n) = A032741(n).

A335543 Numbers with an equal number of deficient and abundant divisors.

Original entry on oeis.org

144, 324, 336, 756, 900, 1176, 1848, 2100, 2184, 2940, 3200, 3520, 4000, 4160, 4400, 5200, 5952, 10880, 11440, 12160, 12348, 12544, 13600, 14720, 15200, 16368, 17360, 18304, 18400, 18560, 19344, 19360, 19404, 22932, 23200, 27040, 28600, 29988, 33516, 40572, 47124

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

This sequence is infinite. For example, 3200*p is a term for all primes p >= 257.

The least odd term of this sequence is a(1273824) = 3010132125.

			144 is a term since it has 7 deficient divisors: {1, 2, 3, 4, 8, 9, 16} and 7 abundant divisors: {12, 18, 24, 36, 48, 72, 144}.

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

Cf. A000203, A005100, A005101, A080224, A080226, A335544.

ab[n_] := DivisorSigma[1, n] - 2n; eqdivQ[n_] := Count[(abs = ab/@Divisors[n]), ?(# > 0 &)] == Count[abs, ?(# < 0 &)]; Select[Range[50000], eqdivQ]

Numbers k such that A080224(k) = A080226(k).

A335544 Numbers with more abundant divisors than deficient divisors.

Original entry on oeis.org

216, 240, 288, 360, 432, 480, 504, 540, 576, 600, 648, 672, 720, 792, 840, 864, 936, 960, 972, 1008, 1056, 1080, 1120, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1728, 1800, 1872, 1920, 1944, 2016, 2112, 2160, 2240, 2268, 2304

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

This sequence is infinite. For example, 216*p is a term for all primes p.
The least odd term of this sequence is a(16317321) = 638512875.
Apparently, this sequence has an asymptotic density of about 0.025.

			216 is a term since it has 8 abundant divisors, {12, 18, 24, 36, 54, 72, 108, 216}, and only 7 deficient divisors, {1, 2, 3, 4, 8, 9, 27}.

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

Cf. A000203, A005100, A005101, A080224, A080226, A335543.

ab[n_] := DivisorSigma[1, n] - 2n; moreAbQ[n_] := Count[(abs = ab/@Divisors[n]), ?(# > 0 &)] > Count[abs, ?(# < 0 &)]; Select[Range[50000], moreAbQ]

Numbers k such that A080224(k) > A080226(k).

A357460 Numbers whose number of deficient divisors is equal to their number of nondeficient divisors.

Original entry on oeis.org

72, 108, 120, 168, 180, 252, 420, 528, 560, 624, 1188, 1224, 1368, 1400, 1404, 1632, 1656, 1824, 1836, 1960, 1980, 2040, 2052, 2088, 2208, 2232, 2280, 2340, 2484, 2664, 2760, 2772, 2784, 2856, 2952, 2976, 3060, 3096, 3132, 3192, 3200, 3276, 3348, 3384, 3420, 3432

Offset: 1

Amiram Eldar, Sep 29 2022

nonn

Numbers k such that A080226(k) = A341620(k).
This sequence is infinite: if p >= 17 is a prime then 72*p is a term.
The least odd term of this sequence is a(36126824) = A357461(1) = 3010132125.
Since the number of divisors of any term is even, none of the terms are squares.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 10, 131, 1172, 12003, 120647, 1199147, 11992293, 120089446, ... . Apparently, the asymptotic density of this sequence exists and is equal to about 0.012.

			72 is a term since it has 12 divisors, 6 of them (1, 2, 3, 4, 8 and 9) are deficient and 6 (6, 12, 18, 24, 36 and 72) are not.

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

Subsequence of A000037 and A005101.

Cf. A080226, A335543, A335544, A341620, A357461, A357462.

q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, 1, -1] &] == 0; Select[Range[3500], q]

is(n) = sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;

A335542 Numbers with a record number of deficient divisors.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 60, 90, 150, 210, 315, 630, 990, 1575, 1890, 2310, 3465, 4620, 6930, 11550, 13860, 17325, 20790, 30030, 39270, 45045, 60060, 78540, 90090, 117810, 131670, 180180, 196350, 219450, 225225, 255255, 270270, 353430, 395010, 450450, 510510, 746130

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

The corresponding numbers of deficient divisors are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 17, 18, 22, ...

			2 is in the sequence since it is the least number with 2 deficient divisors, 1 and 2. The next number with more than 2 deficient divisors is 4, which has 3 deficient divisors, 1, 2, and 4.

Amiram Eldar, Table of n, a(n) for n = 1..120

Cf. A000203, A005100, A080226, A302934, A335540.

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
Module[{nn=800000,lst},lst=Table[{n,Count[Divisors[n],?(DivisorSigma[1,#]<2#&)]},{n,nn}];DeleteDuplicates[lst,GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* _Harvey P. Dale, May 06 2023 *)

Numbers m such that A080226(m) > A080226(k) for all k < m.

cp's OEIS Frontend

A080224 Number of abundant divisors of n.

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A187793 Sum of the deficient divisors of n.

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A080225 Number of perfect divisors of n.

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A341620 Number of nondeficient divisors of n.

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A294926 Number of proper divisors of n that are deficient (A005100).

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A335543 Numbers with an equal number of deficient and abundant divisors.

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A335544 Numbers with more abundant divisors than deficient divisors.

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