A211779 -id:A211779 - cp's OEIS frontend

A066218 Numbers k such that sigma(k) = Sum_{j|k, j

198, 608, 11322, 20826, 56608, 3055150, 565344850, 579667086, 907521650, 8582999958, 13876688358, 19244570848, 195485816050, 255701999358, 1038635009650, 1410759512050, 3308222326688, 6293446033554, 12859914783762, 15343909268584, 18359652610976, 19142664182226, 41584649258178, 45090324794034, 56293124233554

Offset: 1

Joseph L. Pe, Dec 17 2001

I propose this generalization of perfect numbers: for an arithmetical function f, the "f-perfect numbers" are the n such that f(n) = sum of f(k) where k ranges over proper divisors of n. The usual perfect numbers are i-perfect numbers, where i is the identity function. This sequence lists the sigma-perfect numbers. It is not hard to see that the EulerPhi-perfect numbers are the powers of 2 and the d-perfect numbers are the squares of primes (d(n) denotes the number of divisors of n).
Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.
a(17) > 2*10^12. - Giovanni Resta, Jun 20 2013
Numbers k such that A296075(k) = 0. - Amiram Eldar, Apr 16 2024
No more terms < 10^14. - Jud McCranie, Nov 28 2024

			Proper divisors of 198 = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99}; sum of their sigma values = 1 + 3 + 4 + 12 + 13 + 12 + 39 + 36 + 48 + 144 + 156 = 468 = sigma(198).

Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Giovanni Resta, 34 numbers > 3*10^12 which belong to the sequence.

Cf. A211779, A000203, A066229, A066230, A296075.

f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

is(n)=sumdiv(n,d,sigma(d))==2*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Integer n = p1^k1 * p2^k2 * ... * pm^km is in this sequence if and only if g(p1^k1)*g(p2^k2)*...*g(pm^km)=2, where g(p^k) = (p^(k+2)-(k+2)*p+k+1)/(p^(k+1)-1)/(p-1) for prime p and integer k>=1. - Max Alekseyev, Oct 23 2008

More terms from Naohiro Nomoto, May 07 2002
a(7)-a(8) from Farideh Firoozbakht, Sep 18 2006
a(9)-a(13) from Donovan Johnson, Jun 25 2012
a(14)-a(16) from Giovanni Resta, Jun 20 2013
a(17)-a(25) from Jud McCranie, Nov 28 2024

A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 10, 1, 30, 1, 12, 11, 32, 1, 36, 1, 38, 13, 16, 1, 92, 7, 18, 19, 46, 1, 74, 1, 80, 17, 22, 15, 140, 1, 24, 19, 116, 1, 90, 1, 62, 51, 28, 1, 256, 9, 62, 23, 70, 1, 136, 19, 140, 25, 34, 1, 286, 1, 36, 61, 192, 21, 122, 1, 86, 29, 114

Offset: 1

Paolo P. Lava, Feb 19 2015

nonn

a(n) = 1 if n is prime.

			The aliquot parts of 8 are 1, 2, 4 and their sum is 7.
Now, let us calculate the aliquot parts of 1, 2 and 4:
1 => 0;  2 => 1;  4 => 1, 2.  Their sum is 0 + 1 + 1 + 2 = 4.
Let us calculate the aliquot parts of 1, 1, 2:
1 => 0;  1 = > 0; 2 => 1. Their sum is 1.
We have left 1: 1 => 0.
Finally, 7 + 4 + 1 = 12. Therefore a(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)
Jon Maiga, Computer-generated formulas for A255242, Sequence Machine.

Cf. A001047, A001065, A001787, A006516, A016127, A016130, A016135, A255243, A330575.

Sequences that appear in the convolution formulas: A000203, A001065, A051953, A062790, A067824, A074206, A157658, A174725, A174726, A191161, A211779, A245211, A323910, A253249.

with(numtheory): P:=proc(q) local a,b,c,k,n,t,v;
for n from 1 to q do b:=0; a:=sort([op(divisors(n))]); t:=nops(a)-1;
while add(a[k],k=1..t)>0 do b:=b+add(a[k],k=1..t); v:=[];
for k from 2 to t do c:=sort([op(divisors(a[k]))]); v:=[op(v),op(c[1..nops(c)-1])]; od;
a:=v; t:=nops(a); od; print(b); od; end: P(10^3);

f[s_] := Flatten[Most[Divisors[#]] & /@ s]; a[n_] := Total@Flatten[FixedPointList[ f, {n}]] - n; Array[a, 100] (* Amiram Eldar, Apr 06 2019 *)

ali(n) = setminus(divisors(n), Set(n));
a(n) = my(list = List(), v = [n]); while (#v, my(w = []); for (i=1, #v, my(s=ali(v[i])); for (j=1, #s, w = concat(w, s[j]); listput(list, s[j]));); v = w;); vecsum(Vec(list)); \\ Michel Marcus, Jul 15 2023

a(1) = 0.
a(2^k) = k*2^(k-1) = A001787(k), for k>=1.
a(n^k) = (n^k-2^k)/(n-2), for n odd prime and k>=1.
In particular:
a(3^k)  = A001047(k-1);
a(5^k)  = A016127(k-1);
a(7^k)  = A016130(k-1);
a(11^k) = A016135(k-1).
From Antti Karttunen, Nov 22 2024: (Start)
a(n) = A330575(n) - n.
Also, following formulas were conjectured by Sequence Machine:
a(n) = (A191161(n)-n)/2.
a(n) = Sum_{d|n} A001065(d)*A074206(n/d). [Compare to David A. Corneth's Apr 13 2020 formula for A330575]
a(n) = Sum_{d|n} A051953(d)*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A174726(n/d).
a(n) = Sum_{d|n} A062790(d)*A253249(n/d).
a(n) = Sum_{d|n} A157658(d)*A191161(n/d).
a(n) = Sum_{d|n} A174725(d)*A211779(n/d).
a(n) = Sum_{d|n} A245211(d)*A323910(n/d).
(End)

A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

Original entry on oeis.org

0, 2, 2, 7, 2, 14, 2, 18, 9, 18, 2, 43, 2, 22, 20, 41, 2, 54, 2, 57, 24, 30, 2, 106, 13, 34, 31, 71, 2, 110, 2, 88, 32, 42, 28, 162, 2, 46, 36, 142, 2, 138, 2, 99, 81, 54, 2, 237, 17, 102, 44, 113, 2, 178, 36, 178, 48, 66, 2, 325, 2, 70, 99, 183, 40, 194, 2

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Numbers n such that n divides a(n) are given in A068978.

			For n = 12: Sum_{d|n, d<n} d * tau(n / d) = 1*6 + 2*4 + 3*3 + 4*2 + 6*2 = 43.

Antti Karttunen, Table of n, a(n) for n = 1..27144 (first 1000 terms from Jaroslav Krizek)

Cf. A000005, A000203, A007429, A068978, A098198, A211779.

Table[Sum[d*DivisorSigma[0, n/d], {d, Most[Divisors[n]]}], {n, 100}] (* T. D. Noe, Apr 27 2012 *)

A211780(n) = sumdiv(n, d, sigma(d))-n; \\ Antti Karttunen, Nov 13 2017

A211780=lambda n:sum(sigma(d) for d in divisors(n, generator=True))-n
from sympy import divisor_sigma as sigma, divisors # M. F. Hasler, Jun 03 2024

a(n) = A007429(n) - n = A211779(n) + A000203(n) - n .
a(n) = (Sum_{d|n} A000203(d)) - n. - Antti Karttunen, Nov 13 2017
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^4/36 - 1 = 1.705808... . - Amiram Eldar, Jun 06 2024

Name edited by M. F. Hasler, Jun 03 2024

A318445 a(n) = Sum_{d|n, dA005187(d).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 11, 5, 12, 1, 25, 1, 15, 13, 26, 1, 34, 1, 37, 16, 23, 1, 62, 9, 27, 21, 47, 1, 70, 1, 57, 24, 36, 20, 97, 1, 39, 28, 90, 1, 93, 1, 71, 55, 46, 1, 139, 12, 77, 37, 83, 1, 118, 28, 115, 40, 58, 1, 193, 1, 61, 71, 120, 32, 142, 1, 109, 47, 133, 1, 228, 1, 75, 86, 119, 31, 164, 1, 199, 71, 83, 1, 256, 41

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005187, A293214, A293447, A318446, A318447.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A318445(n) = sumdiv(n,d,(dA005187(d));

a(n) = Sum_{d|n, dA005187(d).
a(n) = A318446(n) - A005187(n).
a(n) = A211779(n) + A318447(n).
a(n) = A293447(A293214(n)).

A318447 a(n) = Sum_{d|n, dA294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 3, 2, 0, 0, 1, 0, 2, 3, 7, 0, -8, 2, 9, 3, 4, 0, 2, 0, 0, 7, 14, 5, -10, 0, 15, 9, -2, 0, 9, 0, 12, 7, 18, 0, -22, 3, 18, 14, 16, 0, 6, 9, 1, 15, 24, 0, -24, 0, 25, 13, 0, 11, 26, 0, 26, 18, 25, 0, -45, 0, 33, 20, 28, 10, 32, 0, -14, 13, 37, 0, -15, 16, 38, 24, 13, 0, -8, 12, 34, 25, 41, 17, -52, 0

Offset: 1

Antti Karttunen, Aug 27 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A005187, A211779, A292257, A294898, A296074, A318445, A318448.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318447(n) = sumdiv(n,d,(dA294898(d));

a(n) = Sum_{d|n, dA294898(d).
a(n) = A318448(n) - A294898(n).
a(n) = A318445(n) - A211779(n).
a(n) = A296074(n) - A292257(n).

A224488 Numbers m such that k*m = Sum_{j|m, j < m} sigma(j), where k >= 1 is an integer.

Original entry on oeis.org

4, 10, 42, 90, 2448, 4680, 27144, 117000, 154498509, 222970077, 11049088050, 63554826816

Offset: 1

Giovanni Resta, Apr 08 2013

a(13) > 10^11.

			The divisors of 10 smaller than 10 are 1, 2 and 5. Since sigma(1) + sigma(2) + sigma(5) = 10, 10 is in the sequence.

Cf. A211779, A221219.

Select[Range[2, 120000], Mod[Total@DivisorSigma[1, Most@Divisors@#], #] == 0 &]

isok(m) = sumdiv(m, d, if (d!=m, sigma(d))) % m == 0; \\ Michel Marcus, Jul 13 2021

a(9)-a(12) and bound on a(13) from Donovan Johnson.

cp's OEIS Frontend

A066218 Numbers k such that sigma(k) = Sum_{j|k, j

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A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

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A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

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A318445 a(n) = Sum_{d|n, dA005187(d).

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A318447 a(n) = Sum_{d|n, dA294898(d), where A294898(d) = A005187(d) - sigma(d).

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A224488 Numbers m such that k*m = Sum_{j|m, j < m} sigma(j), where k >= 1 is an integer.

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