A187793 -id:A187793 - cp's OEIS frontend

A318876 Sum of divisors d of n for which 2*phi(d) > d.

1, 1, 4, 1, 6, 4, 8, 1, 13, 6, 12, 4, 14, 8, 24, 1, 18, 13, 20, 6, 32, 12, 24, 4, 31, 14, 40, 8, 30, 24, 32, 1, 48, 18, 48, 13, 38, 20, 56, 6, 42, 32, 44, 12, 78, 24, 48, 4, 57, 31, 72, 14, 54, 40, 72, 8, 80, 30, 60, 24, 62, 32, 104, 1, 84, 48, 68, 18, 96, 48, 72, 13, 74, 38, 124, 20, 96, 56, 80, 6, 121, 42, 84, 32, 108, 44, 120, 12, 90, 78, 112, 24, 128, 48, 120, 4, 98, 57, 156, 31, 102, 72, 104, 14, 87

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

Not multiplicative as a(3) = 4, a(5) = 6 and a(7) = 8, but a(105) = 87, not a(3)*a(5)*a(7) = 4*6*8 = 192 = A000593(105).

			n = 105 has divisors [1, 3, 5, 7, 15, 21, 35, 105]. When A083254 is applied to them, all others except the last one result a positive number, thus a(105) = 1+3+5+7+15+21+35 = 87.

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

Cf. A000010, A000203, A006519, A083254, A318874, A318877, A318878.
Cf. also A187793.
Differs from A000593 for the first time at n=105, where a(105) = 87, while A000593(105) = 192.

A318876[n_] := DivisorSum[n,  # &, 2*EulerPhi[#] > # &];
Array[A318876, 100] (* Paolo Xausa, Jul 08 2024 *)

A318876(n) = sumdiv(n,d,((2*eulerphi(d))>d)*d);

a(n) = Sum_{d|n} [2*phi(d) > d]*d, where [ ] are the Iverson brackets.

For all n >= 1, a(n) + A318877(n) + 2*(A006519(n)-1) = A000203(n).

A318678 Sum of deficiencies of deficient divisors of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 7, 4, 8, 8, 11, 5, 13, 12, 13, 5, 17, 9, 19, 9, 19, 20, 23, 6, 24, 24, 22, 13, 29, 16, 31, 6, 31, 32, 33, 10, 37, 36, 37, 10, 41, 24, 43, 25, 30, 44, 47, 7, 48, 34, 49, 31, 53, 23, 53, 14, 55, 56, 59, 17, 61, 60, 46, 7, 63, 40, 67, 43, 67, 40, 71, 11, 73, 72, 58, 49, 75, 48, 79, 11, 63, 80, 83, 25, 83, 84, 85, 26, 89, 33, 89, 61

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

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

Cf. A033879, A187793, A296075, A318441, A318442, A318679.

f[n_] := 2 n - DivisorSigma[1, n]; Array[DivisorSum[#, f, f@ # >= 0 &] &, 92] (* Michael De Vlieger, Sep 04 2018 *)

A318678(n) = sumdiv(n,d,d=d+d-sigma(d); (d>0)*d);

a(n) = Sum_{d|n} [A033879(d) > 0]*A033879(d).

a(n) = A296075(n) + A318679(n).

A357462 Numbers whose sum of deficient divisors is equal to their sum of nondeficient divisors.

Original entry on oeis.org

6, 28, 30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 308, 318, 330, 354, 364, 366, 390, 402, 426, 438, 462, 474, 476, 496, 498, 510, 532, 534, 546, 570, 582, 606, 618, 642, 644, 654, 678, 690, 714, 726, 750, 762, 786, 798, 812, 822, 834

Offset: 1

Amiram Eldar, Sep 29 2022

nonn

Numbers k such that A187793(k) = A187794(k) + A187795(k).
All the terms are nondeficient numbers (A023196).
All the perfect numbers (A000396) are terms.
This sequence is infinite: if k = 2^(p-1)*(2^p-1) is an even perfect number and q > 2^p-1 is a prime, then k*q is a term.
Since the total sum of divisors of any term is even, none of the terms are squares or twice squares.
Are there odd terms in this sequence? There are none below 10^10.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 6, 63, 605, 6164, 61291, 614045, 6139193, 61382607, 613861703, ... . Apparently, the asymptotic density of this sequence exists and equals 0.06138... .

			6 is a term since the sum of its deficient divisors, 1 + 2 + 3 is equal to 6, its only nondeficient divisor.
30 is a term since the sum of its deficient divisors, 1 + 2 + 3 + 5 + 10 + 15 = 36 is equal to the sum of its nondeficient divisors, 6 + 30 = 36.

Cf. A187793, A187794, A187795, A335543, A357460.
Subsequence of A023196 and A028983.
A000396 is a subsequence.

q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, #, -#] &] == 0; Select[Range[1000], q]

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

A206773 Sum of nonprime proper divisors (or nonprime aliquot parts) of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 1, 1, 1, 13, 1, 16, 1, 15, 1, 1, 1, 31, 1, 1, 10, 19, 1, 32, 1, 29, 1, 1, 1, 50, 1, 1, 1, 43, 1, 42, 1, 27, 25, 1, 1, 71, 1, 36, 1, 31, 1, 61, 1, 55, 1, 1, 1, 98, 1, 1, 31, 61, 1, 62, 1, 39, 1, 60, 1, 118, 1, 1, 41, 43, 1

Offset: 1

Michel Lagneau, Jan 10 2013

nonn

Sum of nonprime divisors of n that are less than n.
a(n) = 1 if n is prime or semiprime.
Up to 3*10^12, a(n) = n only for n = 42, 1316, and 131080256. In general, if p = 2^k-1 and q = 4^k-2*2^k-1 are two primes, then n = 2^(k-1)*p*q satisfies a(n) = n. This happens for k= 2, 3, 7, and 19, which give the aforementioned values and 3777871569031248714137. This property makes these values terms of A225028. - Giovanni Resta, May 03 2016

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A001065, A105221, A023890, A187793, A001358, A000469, A225028.

with(numtheory):for n from 1  to 100 do:x:=factorset(n):n1:=nops(x):s:=sum('x[i] ', 'i'=1..n1): s1:=sigma(n)-s-n: if type(n,prime)=true then printf(`%d, `,1) else printf(`%d, `,s1):fi:od:

Table[Plus@@Select[Divisors[n],#Giovanni Resta, May 03 2016 *)

a(n) = A001065(n) - A105221(n)

A210510 Numbers not in the range of the sum of deficient divisors function.

Original entry on oeis.org

2, 5, 9, 11, 16, 17, 21, 23, 25, 26, 29, 33, 35, 37, 39, 41, 45, 47, 49, 50, 51, 53, 55, 58, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 82, 83, 85, 87, 88, 89, 91, 95, 99, 100, 101, 103, 107, 109, 111, 113, 115, 116, 117, 119, 125, 129, 131, 136, 137, 141, 143, 145, 146, 147, 149, 155, 157, 159

Offset: 1

Timothy L. Tiffin, Jan 25 2013

nonn

Numbers which do not appear in A187793 or in A274829; that is, there is no integer N whose sum of deficient divisors is equal to a(n) for any n.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A005100, A187793, A274829 (complement).

sdef[n_] := Total@Select[Divisors@n, DivisorSigma[-1, #] < 2 &]; ric[n_, lap_] := Block[{s, m, p = NextPrime@lap}, m = n*p; While[(s = sdef@m) <= Lim, T[[s]] = 1; ric[m, p]; m *= p; While[(s = sdef@m) <= Lim, T[[s]] = 1; ric[m, p]; m *= p]; p = NextPrime@p; m = n*p]]; seqTo[n_] := (Lim = n; T = 0 Range[n]; T[[1]] = 1; ric[1, 1]; Flatten@Position[T, 0]); seqTo[159] (* Giovanni Resta, Feb 20 2013 *)

v=vector(10^7); for(i=1, 10^7, if(sigma(i)Donovan Johnson, Feb 20 2013 */

A274829 Numbers in the range of the sum of deficient divisors function.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 18, 19, 20, 22, 24, 27, 28, 30, 31, 32, 34, 36, 38, 40, 42, 43, 44, 46, 48, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 84, 86, 90, 92, 93, 94, 96, 97, 98, 102, 104, 105, 106, 108, 110, 112, 114, 118, 120, 121, 122, 123, 124, 126, 127, 128, 130, 132, 133, 134, 135, 138, 139, 140, 142, 144, 148, 150, 151, 152, 153, 154, 156, 158

Offset: 1

Timothy L. Tiffin, Jul 07 2016

nonn

Possible values for the sum of deficient divisors of the positive integers, written in ascending order.

Cf. A005100, A187793, A210510 (complement).

Take[#, 100] &@ Union@ Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &], {n, 10^4}] (* Michael De Vlieger, Jul 13 2016 *)

cp's OEIS Frontend

A318876 Sum of divisors d of n for which 2*phi(d) > d.

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A318678 Sum of deficiencies of deficient divisors of n.

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A357462 Numbers whose sum of deficient divisors is equal to their sum of nondeficient divisors.

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A206773 Sum of nonprime proper divisors (or nonprime aliquot parts) of n.

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A210510 Numbers not in the range of the sum of deficient divisors function.

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A274829 Numbers in the range of the sum of deficient divisors function.

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