A125310 -id:A125310 - cp's OEIS frontend

A187793 Sum of the deficient divisors of n.

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

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).

A274338 The period 10 sequence of the iterated sum of deficient divisors function (A187793) starting at 52.

Original entry on oeis.org

52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065).  It appears to be the only one of period (order, length) 10 that A187793 generates under iteration.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.

			a(1) = 52;
a(2) = sigma(52) = 98;
a(3) = sigma(98) = 171;
a(4) = sigma(171) = 260;
a(5) = sigma(260) - 260 - 20 = 308;
a(6) = sigma(308) - 308 - 28 = 336;
a(7) = 1 + 2 + 3 + 4 + 7 + 8 + 14 + 16 + 21 = 76 [since 336 has more abundant divisors than deficient ones];
a(8) = sigma(76) = 140;
a(9) = sigma(140) - 140 - 70 - 28 - 20 = 78;
a(10) = sigma(78) - 78 - 6 = 84;
a(11) = sigma(84) - 84 - 42 - 28 - 12 - 6 = 52 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274339, A274340, A274380, A274549, A275082.

a(n)=n=n%10; if(n>0, sumdiv(a(n-1),d,if(sigma(d,-1)<2,d,0)), 84) \\ Charles R Greathouse IV, Jun 23 2016

Vec(x*(52 + 98*x + 171*x^2 + 260*x^3 + 308*x^4 + 336*x^5 + 76*x^6 + 140*x^7 + 78*x^8 + 84*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^50)) \\ Colin Barker, Jan 30 2020

a(n+10) = a(n).

G.f.: x*(52 + 98*x + 171*x^2 + 260*x^3 + 308*x^4 + 336*x^5 + 76*x^6 + 140*x^7 + 78*x^8 + 84*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Jan 30 2020

A274339 The period 3 sequence of the iterated sum of deficient divisors function (A187793) starting at 15.

Original entry on oeis.org

15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be the only one of period (order, length) 3 that A187793 generates under iteration.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.
A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

			a(1) = 15;
a(2) = sigma(15) = 24;
a(3) = sigma(24) - 24 - 12 - 6 = 18;
a(4) = sigma(18) - 18 - 6 = 15 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274340, A274380, A274549, A275082, A284326.

LinearRecurrence[{0,0,1},{15,24,18},90] (* or *) PadRight[{},90,{15,24,18}] (* Harvey P. Dale, Aug 06 2023 *)

Vec(3*x*(5 + 8*x + 6*x^2) / ((1 - x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Jan 30 2020

a(n+3) = a(n).

G.f.: 3*x*(5 + 8*x + 6*x^2) / ((1 - x)*(1 + x + x^2)). - Colin Barker, Jan 30 2020

A274340 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 19.

Original entry on oeis.org

19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274380.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.

			a(1) = 19;
a(2) = sigma(19) = 20;
a(3) = sigma(20) - 20 = 22;
a(4) = sigma(22) = 36;
a(5) = sigma(36) - 36 - 18 - 12 - 6 = 19 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274380, A274549, A275082.

PadRight[{},100,{19,20,22,36}] (* Paolo Xausa, Oct 16 2023 *)

Vec(x*(19 + 20*x + 22*x^2 + 36*x^3) / (1 - x^4) + O(x^80)) \\ Colin Barker, Jan 30 2020

a(n+4) = a(n).
a(n) = A187793(a(n-1)).
G.f.: x*(19 + 20*x + 22*x^2 + 36*x^3) / (1 - x^4). - Colin Barker, Jan 30 2020

A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

Original entry on oeis.org

34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274340.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.
A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

			a(1) = 34;
a(2) = sigma(34) = 54;
a(3) = sigma(54) - 18 - 6 = 42;
a(4) = sigma(42) - 42 - 6 = 48;
a(5) = sigma(48) - 48 - 24 - 12 - 6 = 34 = a(1);
  :
  :

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274340, A274549, A275082, A284326.

Vec(2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^80)) \\ Colin Barker, Jan 30 2020

a(n+4) = a(n).

G.f.: 2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)). - Colin Barker, Jan 30 2020

A274549 Numbers found in the cycles of the iterated sum of deficient divisors function.

Original entry on oeis.org

1, 6, 15, 18, 19, 20, 22, 24, 28, 34, 36, 42, 48, 52, 54, 76, 78, 84, 90, 98, 140, 171, 260, 308, 336, 496, 8128, 33550336, 8589869056

Offset: 1

Timothy L. Tiffin, Jun 27 2016

The 1-cycles (or fixed points) greater than 1 are given in A125310, the 3-cycle terms are given in A274339, the 4-cycle terms are given in A274340 and A274380, and the 10-cycle terms are given in A274338. This sequence was suggested to me by Michel Marcus when I was submitting the 3-cycle, 4-cycle, and 10-cycle sequences.

Cf. A125310, A187793, A274338, A274339, A274340, A274380.

A198471 Abundant numbers that are smaller than the sum of their deficient divisors.

Original entry on oeis.org

20, 30, 42, 66, 70, 78, 88, 102, 104, 114, 138, 150, 174, 186, 210, 220, 222, 246, 258, 260, 272, 282, 294, 304, 308, 318, 330, 340, 354, 364, 366, 368, 380, 390, 402, 426, 438, 450, 460, 462, 464, 474, 476, 498, 510, 532, 534, 546, 550, 570, 572, 580, 582, 606

Offset: 1

Timothy L. Tiffin, Jan 07 2013

nonn

The primitive abundant number sequence (A071395) is a subsequence of this sequence.

			a(5) = 70 since 70 is smaller than 74, which is the sum of its deficient divisors.

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

Cf. A071395, A198471, A125310.

totdef[n_] := Total@Select[Divisors@n, DivisorSigma[-1, #] < 2 &]; Select[Range[570], DivisorSigma[-1, #] > 2 && # < totdef[#] &] (* Giovanni Resta, Jan 09 2013 *)

isok(n) = (sigma(n) > 2*n) && (n < sumdiv(n, d, if (sigma(d) < 2*d, d))); \\ Michel Marcus, Jun 21 2019

a(21)-a(50) from Giovanni Resta, Jan 09 2013

A294900 Numbers k such that k = sum of nonabundant proper divisors of k (A294888).

Original entry on oeis.org

6, 24, 28, 126, 496, 8128, 5594428, 33550336, 8589869056, 17589794838, 35439846824, 49380301744

Offset: 1

Antti Karttunen, Nov 14 2017

Naturally, all the terms of A000396, including 137438691328, are in this sequence. - Antti Karttunen, Dec 01 2017
Thus, if there are infinitely many Mersenne primes, then this sequence is also, by definition of even perfect numbers, infinite. - Iain Fox, Dec 02 2017
All non-perfect terms are abundant. Proof: Assume d is a deficient number in this sequence. Because multiples of abundant numbers are abundant, d cannot have an abundant divisor, thus all its divisors are nonabundant. Since d is in this sequence, the sum of its proper divisors, which are all nonabundant, must equal d. However, if this were true, then d would be perfect. Therefore, this sequence contains no deficient numbers. - Iain Fox, Dec 07 2017
Questions from Iain Fox, Dec 07 2017: (Start)
Are there an infinite number of abundant terms?
Are all abundant terms in this sequence even?
(End)
No other terms up to 10^10. - Iain Fox, Dec 07 2017
a(13) > 6*10^10. - Giovanni Resta, Dec 11 2017
In comparison, the numbers which are the sum of their abundant proper divisors seems to be scarcer: up to 6*10^10 only 19514300 and 16333377500 have this property. - Giovanni Resta, Dec 11 2017
From Iain Fox, Dec 11 2017: (Start)
The first abundant term without a perfect divisor is 35439846824.
This term and any other abundant terms without perfect divisors are also terms in A125310.
(End)

Index entries for sequences where any odd perfect numbers must occur

Fixed points of A294888.
Subsequence of A005835; A000396 is a subsequence.
Cf. A125310.

isok(n) = sumdiv(n, d, if ((dMichel Marcus, Nov 17 2017

normalize(f)=f=select(v->v[2],f~)~;if(vecmax(matsize(f)),f,factor(1));
is(n,f=factor(n))=
{
my(p=Mat(f[,1]),g,s);
forvec(v=apply(k->[0,k],f[,2]~),
g=normalize(concat(p,v~));
if(sigma(g,-1)<=2,
s+=factorback(g)
);
);
s==if(sigma(f,-1)>2,n,2*n);
}
forfactored(n=6,10^9, if(is(n[1],n[2]), print1(n[1]", "))) \\ Charles R Greathouse IV, Dec 08 2017

a(9) from Iain Fox, Dec 07 2017

a(10)-a(12) from Giovanni Resta, Dec 11 2017

A198470 Numbers that are larger than the sum of their deficient divisors.

Original entry on oeis.org

12, 18, 24, 36, 40, 48, 54, 56, 60, 72, 80, 84, 96, 100, 108, 112, 120, 126, 132, 140, 144, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 216, 224, 228, 234, 240, 252, 264, 270, 276, 280, 288, 300, 306, 312, 320, 324, 336, 342, 348, 350, 352, 360, 372, 378, 384, 392, 396, 400, 408, 414

Offset: 1

Timothy L. Tiffin, Jan 07 2013

nonn

This sequence is a subsequence of the abundant numbers A005101.

Includes 2^m*p if p is an odd prime and m >= ceiling(log_2(p+1))-1. - Robert Israel, Dec 28 2017

			a(4) = 36 since 36 is larger than 19, which is the sum of its deficient divisors.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A125310.

filter:= n -> convert(select(d -> numtheory:-sigma(d) < 2*d, numtheory:-divisors(n)),`+`)Robert Israel, Dec 28 2017

totdef[n_] := Total@Select[Divisors@n, DivisorSigma[-1, #] < 2 &];
Select[Range[300], DivisorSigma[-1, #] > 2 && # > totdef[#] &] (* Giovanni Resta, Jan 09 2013 *)

is_A198470(n)=!fordiv(n,d,sigma(d)<2*d & (n-=d)<=0 & return)  \\ M. F. Hasler, Jan 11 2013

More terms from Robert Israel, Dec 28 2017

cp's OEIS Frontend

A187793 Sum of the deficient divisors of n.

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A294886 Sum of deficient proper divisors of n.

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A274338 The period 10 sequence of the iterated sum of deficient divisors function (A187793) starting at 52.

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A274339 The period 3 sequence of the iterated sum of deficient divisors function (A187793) starting at 15.

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A274340 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 19.

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A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

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A274549 Numbers found in the cycles of the iterated sum of deficient divisors function.

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