A294889 -id:A294889 - cp's OEIS frontend

A187795 Sum of the abundant divisors of n.

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)

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).

A294886 Sum of deficient proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 22, 11, 14, 1, 18, 6, 16, 13, 28, 1, 36, 1, 31, 15, 20, 13, 19, 1, 22, 17, 30, 1, 48, 1, 40, 33, 26, 1, 34, 8, 43, 21, 46, 1, 42, 17, 36, 23, 32, 1, 40, 1, 34, 41, 63, 19, 72, 1, 58, 27, 74, 1, 27, 1, 40, 49, 64, 19, 84, 1, 46, 40, 44, 1, 52, 23, 46, 33, 92, 1, 90, 21

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Sum of divisors of n smaller than n that are deficient numbers (in A005100).

			Proper divisors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45. Of these 1, 2, 3, 5, 9, 10, 15 and 45 are in A005100, thus a(90) = 1+2+3+5+9+10+15+45 = 90.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A001065, A005100, A001065, A187793, A294926, A294934, A294887, A294888, A294889.

Cf. A125310 (fixed points).

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

PARI
```
A294886(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA294934(d)*d.
a(n) = A187793(n) - (A294934(n)*n).
a(n) + A294887(n) = A001065(n).

A294888 Sum of nonabundant proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 24, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 25, 1, 22, 17, 30, 1, 54, 1, 40, 33, 26, 1, 40, 8, 43, 21, 46, 1, 48, 17, 64, 23, 32, 1, 46, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 33, 1, 40, 49, 64, 19, 90, 1, 46, 40, 44, 1, 86, 23, 46, 33, 92, 1, 96, 21

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Sum of divisors of n smaller than n that are nonabundant numbers (in A263837).

			Proper divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45. Of these 1, 2, 3, 5, 6, 9, 10, 15 and 45 are in A263837, thus a(90) = 1+2+3+5+6+9+10+15+45 = 96.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A001065, A263837, A294928, A294935, A294886, A294887, A294889.

Cf. A294900 (fixed points).

a[n_] := DivisorSum[n, Boole[# < n && DivisorSigma[1, #] <= 2#] * #&];
Array[a, 100] (* Jean-François Alcover, Nov 17 2017 *)

PARI
```
A294888(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA294935(d)*d.

a(n) + A294889(n) = A001065(n).

A294887 Sum of nondeficient proper divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 36, 0, 0, 0, 20, 0, 6, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 24, 0, 28, 0, 0, 0, 68, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 96, 0, 0, 0, 0, 0, 6, 0, 60, 0, 0, 0, 88, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 90, 0, 0, 0, 20, 0, 6, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Sum of divisors n smaller than n that are nondeficient numbers (in A023196).

			Proper divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, and 45. Of these 6, 18 and 30 are in A023196, thus a(90) = 6+18+30 = 54.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors.

Cf. A001065, A023196, A294927, A294936, A294886, A294888, A294889.

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

PARI
```
A294887(n) = sumdiv(n, d, (d=(2*d))*d);
```

a(n) = Sum_{d|n, dA294936(d)*d. - Typo in A-number corrected by Antti Karttunen, Apr 04 2022

a(n) + A294886(n) = A001065(n).

A296074 Sum of deficiencies of the proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 6, 1, 5, 1, 8, 7, 4, 1, 9, 1, 9, 9, 12, 1, 2, 5, 14, 8, 13, 1, 16, 1, 5, 13, 18, 11, 3, 1, 20, 15, 8, 1, 24, 1, 21, 18, 24, 1, -9, 7, 27, 19, 25, 1, 20, 15, 14, 21, 30, 1, -1, 1, 32, 24, 6, 17, 40, 1, 33, 25, 40, 1, -27, 1, 38, 32, 37, 17, 48, 1, -1, 22, 42, 1, 9, 21, 44, 31, 26, 1, 18, 19, 45, 33, 48, 23, -36, 1, 53, 36, 33

Offset: 1

Antti Karttunen, Dec 04 2017

			For n = 6, whose proper divisors are 1, 2, 3, their deficiencies are 1, 1, 2, thus a(6) = 1+1+2 = 4.
For n = 12, whose proper divisors are 1, 2, 3, 4, 6, their deficiencies are 1, 1, 2, 1, 0, thus a(12) = 1+1+2+1+0 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A033879.

Cf. also A294886, A294887, A294888, A294889, A293438 (product of).

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2; a[1] = 0; a[n_] := Module[{f = FactorInteger[n]}, 3 * Times @@ f1 @@@ f - Times @@ f2 @@@ f - 2*n]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A033879(n) = ((2*n)-sigma(n));
A296074(n) = sumdiv(n,d,(dA033879(d));

a(n) = Sum_{d|n, dA033879(d).
a(n) = A296075(n) - A033879(n).
Sum_{k=1..n} a(k) ~ (Pi^2/4 - Pi^4/72 - 1) * n^2. - Amiram Eldar, Dec 04 2023

A254880 Let 's' denote the sum of the abundant numbers in the aliquot parts of x. Sequence lists numbers x such that sigma(s)-s is equal to x.

Original entry on oeis.org

4240, 75640, 193720, 259120, 327104, 669520, 1385480, 1613240, 2231240, 4185472, 12228352, 26373640, 35095456, 37497520, 45085240, 48211120, 62156512, 64754272, 81263920, 82228432, 84099808, 109455424, 111330208, 118899616, 118988440, 129663880, 137013536, 139367320

Offset: 1

Paolo P. Lava, Feb 10 2015

nonn

			Aliquot divisors of 4240 are 1, 2, 4, 5, 8, 10, 16, 20, 40, 53, 80, 106, 212, 265, 424, 530, 848, 1060, 2120 and the abundant numbers are 20, 40, 80, 1060, 2120. Their sum is 3320 and sigma(3320) - 3320 = 4240.

Cf. A001065, A005101, A254878, A254879, A294889.

with(numtheory); P:=proc(q) local a,b,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]); b:=0;
for k from 1 to nops(a)-1 do if sigma(a[k])>2*a[k]
then b:=b+a[k]; fi; od; if sigma(b)-b=n
then print(n); fi; od; end: P(10^9);

seqQ[n_] := Module[{s = Total@Select[Most[Divisors[n]], DivisorSigma[1,#] > 2# &]}, s>0 && DivisorSigma[1,s] - s == n]; Select[Range[10^6], seqQ] (* Amiram Eldar, Mar 20 2019 *)

A001065(A294889(a(n))) = a(n).

a(3) inserted and a(11)-a(28) added by Amiram Eldar, Mar 20 2019

cp's OEIS Frontend

A187795 Sum of the abundant divisors of n.

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

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A294886 Sum of deficient proper divisors of n.

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A294888 Sum of nonabundant proper divisors of n.

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A294887 Sum of nondeficient proper divisors of n.

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A296074 Sum of deficiencies of the proper divisors of n.

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A254880 Let 's' denote the sum of the abundant numbers in the aliquot parts of x. Sequence lists numbers x such that sigma(s)-s is equal to x.

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